Commun. Math. Phys. 257, 1–28 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1341-6
Communications in
Mathematical Physics
A Spin Decomposition of the Verlinde Formulas for Type A Modular Categories Christian Blanchet L.M.A.M., Universit´e de Bretagne-Sud, BP 573, 56017 Vannes, France. E-mail:
[email protected] Received: 20 March 2003 / Accepted: 30 December 2004 Published online: 15 April 2005 – © Springer-Verlag 2005
Abstract: A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU (N ). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.
0. Introduction Given a simple, simply connected complex Lie group G, the Verlinde formula [35] is a combinatorial function VG : (K, g) → VG (K, g) associated with G (here the integers K and g are respectively the level and the genus). In conformal field theory this formula gives the dimension of the so called conformal blocks. Its combinatorics was intensively studied since this formula has a deep interpretation as the rank of a space of generalized theta functions (sections of some bundle over the moduli space of G-bundles over a Riemann surface) [6, 5, 15, 28]. See [8, 9], for a development using methods of symplectic geometry. We will consider here a purely topological approach to Verlinde formulas related with SU (N ). The genus g Verlinde formula associated with a modular category [30] is the dimension of the TQFT-module of a genus g surface; the general formula is given in [30, IV,12.1.2]. Various constructions of modular categories are known, either from quantum groups [2, 4, 29] or from skein theory [34, 11, 7]. The geometric Verlinde formula for the group SU (N ) at level K is recovered from the so called SU (N, K) modular category. This modular category can be obtained either from the quantum group Uq sl(N ) when
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q = s 2 is a primitive (N + K)th root of unity or from Homfly skein theory. Its simple objects correspond to the weights in the fundamental alcove. One may also consider a modular category with less simple objects. This was done for gcd(N, K) = 1 by restricting to representations whose heighest weight is in the root lattice, and was called the projective or P SU (N) theory [17, 36, 22, 18, 19]. Using an appropriate choice of the framing parameter in Homfly skein theory, we have obtained in [11] a variant which is defined for all N, K. We are not aware of a quantum group approach to these reduced modular categories for gcd(N, K) > 1. Nevertheless we find it convenient to call them P U (N, K) modular categories. In our construction the simple object corresponding to the deformation of the determinant of the vector representation of sl(N ) may be non-trivial; we think that a version of the quantum group Uq (gl(N )) could be used here. As is well known, the Verlinde formula for the SU (N, K) modular category coincides with the formula in conformal field theory for the group SU (N ); dN,K (g) = VSU (N) (K, g). We show that for the P U (N, K) modular category the Verlinde formula is dN,K (g) , d˜N,K (g) = N g where N =
N gcd(N,K) .
These integral numbers satisfy the level-rank duality relation d˜N,K (g) = d˜K,N (g) ,
which is an integral version of a reciprocity formula in [26] (see also [17]). Our main contribution here is to show that under certain conditions the TQFT modules decompose in blocks indexed by spin type structures (respectively 1-dimensional cohomology classes) on the surface, and compute the corresponding refined Verlinde formulas. An important part of this paper is devoted to the spin decomposition theorem. The proof is given in the general case of a modulo d spin modular category; this notion, developed in Sect. 2, is new and appears in the Z/d graded cases which are not weakly non-degenerate in [19]. As a motivation, we give below the combinatorial counterpart of this theorem for the A series, in the special case where the rank is even and divides the level (Theorem 4.5). We consider the action of Z/N on the set N,K = {λ = (λ1 , . . . , λN ), K ≥ λ1 ≥ · · · ≥ λN−1 ≥ λN = 0} , given for the generator of the cyclic group Z/N by (λ1 , . . . , λN−1 , 0) −→ (K, λ1 , . . . , λN−1 ) − (λN−1 , . . . , λN−1 ) . We denote by orb(λ) the cardinality of the orbit of λ, and by Stab(λ) the stabilizer subgroup. For a, b ∈ Z/N , the numbers λ (a, b) ∈ {0, 1, − 21 , 21 } are defined as follows: If orb(λ) is even, then λ (a, b) =
1 if a and b are zero modulo |Stab(λ)|, 0 otherwise.
Spin Verlinde Formulas
3
If orb(λ) is odd, then 2b 2a 1 |Stab(λ)| |Stab(λ)| if a and b are zero modulo λ (a, b) = 2 (−1) 0 otherwise.
|Stab(λ)| , 2
Theorem 0.1. Suppose that N is even, and that K/N is an odd integer. a) For (a, b) ∈ (Z/N )g × (Z/N )g , the formula g−1 (a,b) dN,K (g) = (N + K)N−1 N ×
1≤i<j ≤N
g λ (aν , bν ) ( orb(λ))2 λ∈N,K ν=1 2−2g π 2 sin (λi − i − λj + j ) N +K
(a,b)
defines a natural number dN,K (g), b) There exists a splitting of the SU (N ) Verlinde formula at level K, (a,b) VSU (N) (K, g) = dN,K (g) . (a,b)∈(Z/N)g ×(Z/N)g
For N = 2, the spin TQFT producing the above decomposition was studied in [13], and a nice algebraic-geometric interpretation was obtained by Andersen and Masbaum [1]. We quote that for N > 2, the involved spin structures are not the usual ones. These structures have coefficients modulo an even integer; they can be understood as something intermediate between the usual spin structures (with modulo 2 coefficients) and complex spin structures. The convenient formalism for the TQFT involving these structures should be a slightly extended version of Homotopy Quantum Field Theory as developed by Turaev [31, 32]. The paper is organized as follows. In Sect. 1 we study spin structures modulo an even integer. In Sect. 2 we define our spin modular categories. In Sect. 3 we establish the spin decomposition of the TQFT in a general context. In Sect. 4 we consider Verlinde formulas for modular categories of the A series. In Sect. 5 we establish similar decomposition theorems based on 1-dimensional cohomology classes. In Sect. 6 we give computer results for small values of N and K. 1. Spin Structures Modulo an Even Integer Let d be an even integer. We recall here the topological definition for modulo d spin structures that was given in [10, 11]. There exists, up to homotopy, a unique non-trivial map g from the classifying space BSO to the Eilenberg-MacLane space K(Z/d, 2). Define the fibration πd : BSpin(Z/d) → BSO to be the pull-back, using g, of the path fibration over K(Z/d, 2). The space BSpin(Z/d) is a classifying space for the non-trivial central extension of the Lie group SO by Z/d, which we denote by Spin(Z/d). For d = 2, this group Spin(Z/2) = Spin is the universal cover of SO, and for general d, we have Spin(Z/d) =
Spin × Z/d . (−1, d/2)
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Now we can use the fibration πd to define structures. Let ESpin(Z/d) = πd∗ (ESO ) be the pull-back of the canonical vector bundle over BSO. Definition 1.1. A modulo d spin structure (or Spin(Z/d) structure) on a manifold M is an homotopy class of fiber maps from the stable tangent bundle τM to ESpin(Z/d) . If non-empty the set of these structures, denoted by Spin(M; Z/d), is affinely isomorphic to H 1 (M; Z/d), by obstruction theory. Moreover the obstruction for existence is a class w2 (M; Z/d) ∈ H 2 (M; Z/d), which is the image of the Stiefel-Whitney class w2 (M) under the homomorphism induced by the inclusion of coefficients Z/2 → Z/d. The Stiefel-Whitney class w2 (M) is zero for every compact oriented manifold whose dimension is lower than or equal to 3, hence spin structures modulo d exist on these manifolds. The various descriptions of the usual spin structures [23] apply to modulo d spin structures. The above definition defines, up to equivalence, a Spin(Z/d) principal bundle over the stable oriented framed bundle P T M (with fiber SO) whose restriction to the fiber is equivalent to the cover map Spin(Z/d) → SO. The cover of P T M defined by the modulo d spin structure is classified by a cohomology class σ ∈ H 1 (P T M, Z/d) whose restriction to the fiber is non-trivial. The above correspondence is one to one; this gives an alternative definition, and we will identify Spin(M; Z/d) with the corresponding affine sub-space of H 1 (P T M, Z/d). Definition 1.2. (Alternative definition of modulo d spin structures) A modulo d spin structure on an oriented manifold M is a cohomology class σ ∈ H 1 (P T M, Z/d) whose restriction to the fiber is non-trivial. Observe that a spin structure can be evaluated on a framed 1-cycle in the manifold. Let us consider an oriented surface . An immersed curve has a preferred framing defined by using the tangent vector. If a closed embedded curve γ bounds a disc, then the evaluation of a modulo d spin structure on the corresponding framed 1-cycle γ˜ is d 2 . Following [3, 16], we get the theorem below which gives a convenient description of modulo d spin structures on the oriented surface . Theorem 1.1. a) Let γ denote an embedded closed curve with γ components. The assignment γ → σ (γ˜ )+(γ ) d2 extends to a well defined map qσ : H1 ( , Z/d) → Z/d. b) The map σ → qσ defines a canonical bijection between Spin( , Z/d) and the set of maps q : H1 ( , Z/d) → Z/d such that for all x, y one has q(x + y) = q(x) + q(y) +
d x.y . 2
(1)
Here . denotes the intersection form on H1 ( , Z/d). Proof. The formula σ (γ˜ ) + (γ ) d2 is unchanged if we add to or remove from the embedded curve γ a trivial component. Let us denote by γ (resp. γ ) the left-handed (resp. right-handed) curve in the band move represented in Fig. 1. We have that γ −γ = ±1. By considering the Gauss map, we see that the cycle γ˜ − γ˜ is homologous in P T
to u, ˜ where u bounds a disc on the surface. We get that the formula is also unchanged under this band move. We deduce that homologous curves in give the same result; hence we have that qσ is well defined on H1 ( , Z). Let γ be a generic immersed curve. Smoothing a crossing changes γ by ±1 and does not change the 1-cycle γ˜ . Hence one has qσ ([γ ]) = σ (γ˜ ) + (γ + I (γ )) d2 , where
Spin Verlinde Formulas
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←→
Fig. 1. Band move
I (γ ) is the number of double points. It follows that for any x, y ∈ H1 ( , Z), Property (1) holds. We deduce that qσ is well defined on H1 ( , Z/d). Bijectivity is established by using that the map qσ commutes with the action of H 1 ( , Z/d).
Let M = S3 (L) be obtained by surgery on the framed link L in the 3-sphere. We want to give a combinatorial description for modulo d spin structures on M. Recall that M is the boundary of a 4-manifold WL called the trace of the surgery. To each σ ∈ Spin(M; Z/d) is associated a relative obstruction w2 (σ ; Z/d) in H 2 (WL , M; Z/d). The group H 2 (WL , M; Z/d) is a free Z/d module of rank m = L. Taking the coordinates of the relative obstruction in the preferred basis (the basis which is Poincar´e dual to the cores of the handles), we get a map ψL : Spin(M; Z/d) → (Z/d)m . The following theorem is proved in [11]. Here BL = (bij ) is the linking matrix. Theorem 1.2. The map ψL : Spin(M; Z/d) → (Z/d)m is injective, and its image is the set of those (c1 , . . . , cm ) which are solutions of the following Z/d-characteristic equation: c1 b11 d BL ... = ... (mod d) . 2 cm bmm 2. Spin Modular Categories A ribbon category is a category equipped with tensor product, braiding, twist and duality satisfying compatibility conditions [30]. If we are given a ribbon category C, then we can define an invariant of links whose components are colored with objects of C. This invariant extends to a representation of the C-colored tangle category and more generally to a representation of the category of C-colored ribbon graphs [30, I.2.5]. In a ribbon category there is a notion of trace of morphisms and dimension of objects. The trace of a morphism f is denoted by f .
f =
f
The dimension of an object V is the trace of the identity morphism 1 V ; we will use the notation V as well as 1 V . We often say quantum trace and dimension to distinguish from the usual trace and dimension in vector spaces. Let k be a field. A ribbon category is said to be k-additive if the Hom sets are k-vector spaces, composition and tensor product are bilinear, and End(trivial object) = k.
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We first recall the definition of a modular category [30, 27]. A modular category over k is a k-additive ribbon category in which there exists a finite family of simple objects λ (here simple means that u → u1λ from k = End(trivial object) to End(λ) is an isomorphism) satisfying the axioms below: • (Domination axiom) For any object V in the category there exists a finite decomposition 1V = i fi 1λi gi , with λi ∈ for every i. • (Non-degeneracy axiom) The following matrix is invertible. S = (Sλµ )λ,µ∈ , where Sλµ ∈ k is the endomorphism of the trivial object associated with the (λ, µ)colored, 0-framed Hopf link with linking +1. It follows that is a representative set of isomorphism classes of simple objects; note that the trivial object is simple, so that we may suppose that is in . If we replace the non-degeneracy axiom by the non-singularity condition below then we have the definition of a pre-modular category (a morphism f ∈ H om(V , W ) is called negligible if for any g ∈ H om(W, V ) we have f g = 0): • (Non-singularity) The category has no non-trivial negligible morphism. A general modularization procedure for pre-modular categories, and a criterion for existence are developed by Brugui`eres [14], and by M¨uger in the context of ∗-categories [24]. Note that after quotienting by negligible morphisms we get the non-singularity condition. This property gives that the pairing H om(V , W ) ⊗ H om(W, V ) → k f ⊗g → f g is non-singular. We can deduce that there exists no non-trivial morphism between nonisomorphic simple objects. One may ask further that the category has direct sums. In fact direct sums may be added in a formal way, and a pre-modular category with direct sums is abelian. This latter fact was pointed out to us by Brugui`eres. In a modular category C, with representative set of simple objects , the Kirby color = λ∈ λ λ is used to define an invariant of closed oriented manifolds with colored graph. If M = S 3 (L) is obtained by surgery on the framed link L in the sphere and contains a colored graph K, then a formula for this invariant is τC (M, K) =
L(, . . . , ), K . U1 () b+ U−1 () b−
Here b+ (resp. b− ) is the number of positive (resp. negative) eigenvalues of the linking matrix BL , and U±1 denotes the unknot with framing ±1. Modular G-categories, with G a group have been introduced by Turaev in [32]; details in the case of an abelian group G, and examples derived from quantum groups are given in [19]. Let G be an abelian group. A G grading of a k-additive monoidal category C is a family of full sub-categories Cj , j ∈ G, such that (i) for any pair of objects V ∈ Obj (Cj ), V ∈ Obj (Cj ), one has V ⊗V ∈ Obj (Cj +j );
Spin Verlinde Formulas
7
(ii) if for some pair of objects V ∈ Obj (Cj ), V ∈ Obj (Cj ), one has H omC (V , V ) = {0}, j = j ; (iii) each object of C is either in ∪j Ob(Cj ), or a direct sum of objects in ∪j Obj (Cj ). Axiom (iii) asks that every object splits as a direct sum of homogeneous objects. Axiom (i) asks that the tensor product is homogeneous, and axiom (ii) that any non-zero morphism with source or target an homogeneous object is homogeneous; this implies that the dual of an homogeneous object has opposite grading. Let C be a modular category. We denote by U(C) the abelian group of isomorphism classes of invertible objects in C (the law is tensor product). If U is a subgroup of U(C) and G = Uˆ is the group of characters χ : U → k∗ , then the category C is G graded. A simple object λ is an object in Cχ if and only if for every J ∈ U equality in Fig. 2 holds. A modular G-category [19] over k is a G graded k-additive ribbon category (C; Cj , j ∈ G) in which there exist finite families j ⊂ Ob(Cj ), j ∈ G, of simple objects λ satisfying the axioms below. • (Domination axiom) For any object V in Cj there exists a finite decomposition 1V = i fi 1λi gi , with λi ∈ j for every i. • (Non-degeneracy axiom) The following matrix is invertible. S = (Sλµ )λ,µ∈0 , where Sλµ ∈ k is the endomorphism of the trivial object associated with the (λ, µ)colored, 0-framed Hopf link with linking +1. It is shown in [32] that a modular G-category with G an abelian group gives invariants of 3-manifolds equipped with a 1-dimensional cohomology class. A modular G-category may not be a modular category, even in the case where G is a finite abelian group (see [19, Sect. 1.6]). We point out that a modular category with a G grading is not necessarily a modular G-category. The reason is that the S-matrix restricted to zero graded objects may be non-invertible. In addition, the zero graded subcategory may be non-modularizable, so that there is no hope to get a modular G-category by using some modularization procedure. The latter fact implies that the modular G-category is not weakly non-degenerate [19]; it is verified for the class of modular categories we consider below. These categories have a Z/d grading with d even and give invariants of 3-manifolds equipped with modulo d spin structures; the relevant version of Homotopy Quantum Field Theories as considered by Turaev, should be understood in relation with [32, Remark 7.4.6].
J
=
λ
χ(J )
λ Fig. 2.
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= θλ
λ
λ
Fig. 3.
For a simple object λ the twist coefficient θλ is defined by Fig. 3. In the quantum group context, this coefficient is given by the action of the so-called quantum Casimir. Definition 2.1. Let d be an even integer (resp. an integer). A modular category is modulo d spin (resp. modulo d cohomological) if it is equipped with an invertible object whose order is d and whose twist coefficient is θ = −1 (resp. θ = 1). In the following we will mainly discuss the spin case; the cohomological case will be considered in Sect. 5. Let (C, ) be a modulo d spin modular category, with as a representative set of simple objects. The object d is isomorphic to the trivial, hence we have d = 1. The dual objects and d−1 have the same quantum dimension. We deduce that d = = ±1. Note that invertible objects are simple, hence the braiding for 2 is identity up to a scalar. By closing we get this scalar and establish the following identity.
= −d
(2)
The next identity is obtained in a similar way.
= d
(3)
It is convenient to fix a primitive d th root of unity ζ , and to identify the group of characters χ : {j , j ∈ Z/d} → k∗ with Z/d. Then the category C is Z/d graded. A simple object V has degree equal to j if and only if the equality in Fig. 4 holds. The Kirby color decomposes according to this grading: = λ λ = j . j ∈Z/d
λ∈
Here the notation λ is the quantum dimension of λ.
= ζj
V
Fig. 4.
V
Spin Verlinde Formulas
9
Ω j’
Ω j’−j
νj
νj Fig. 5. Graded sliding property
The proof of the theorem below is the same as in the ungraded case (see e.g. [7]). The statement holds for any G graded pre-modular category with G an abelian group [19, Prop. 1.4]. Theorem 2.1 (Graded sliding property). Suppose that Vj is an object in Cj , then the equality in Fig. 5 holds for any j ∈ Z/d. Here the framed knot labeled with j may be knotted or linked with the other component labeled Vj ; this fact is represented by the dashed part in the figure. The following theorem is proved from the graded sliding property as was done in [11, Theorem 4.2]. We suppose that C is a modulo d spin modular category, and that =
λ λ = j j ∈Z/d
λ∈
is the graded decomposition of the Kirby color; note [11, Lemma 4.5] that U±1 () = U±1 (d/2 ) . Theorem 2.2. Let C be a modulo d spin modular category, and = j ∈Z/d j be the graded decomposition of the Kirby element. Provided c = (c1 , . . . , cm ) satisfies the modulo d characteristic condition, the formula spin
τC (M, σ ) =
L(c1 , . . . , cm ) U1 () b+ U−1 () b−
defines an invariant of the surgered manifold M = S3 (L) equipped with the modulo d spin structure σ = ψL−1 (c1 , . . . , cm ) . Moreover, ∀M τC (M) =
σ ∈Spin(M;Z/d)
spin
τC (M, σ ).
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3. The Spin Decomposition of the Verlinde Formula If we are given a modular category C then we get a TQFT. In brief we have a functor VC from a cobordism category in dimension 3 to vector spaces. If C is a modulo d spin modular category, then we will construct here a decomposition of the TQFT modules VC ( g ) of a genus g surface and compute the ranks of the summands. The TQFT gives a normalized invariant for a closed 3-manifold M equipped with p1 -structure or 2-framing α and colored graph K. We extend the scalar field k if necesU1 () sary, and fix κ such that κ 6 = U . Let D = κ −3 U1 () = κ 3 U−1 () ; note that −1 () D2 = . The normalized invariant of a connected closed 3-manifold M = (M, α, K) is then [12] ZC (M, α, K) = D−1−b1 (M) κ σ (α) τC (M, K) .
(4)
Here b1 (M) is the first Betti number, and σ (α) is the sigma invariant: σ (α) = 3signature (WL ) − p1 (WL , α), [WL ] , where WL is the trace of the surgery and p1 (WL , α) ∈ H 4 (WL , S 3 (L)) is the relative obstruction to extending α. Let be an oriented surface with structure (a marking [30] or a p1 -structure [12]). We use the object to define a group action on VC ( ) as follows. To an embedded oriented curve γ in we associate the TQFT operator φγ : VC ( ) → VC ( ) corresponding to a trivial cobordism [0, 1] × equipped with a colored link γ 1 (). 2
Here γ 1 is the link 21 × γ equipped with the framing given by the orientation and the 2 normal vector parallel to . The components of this link are colored with . The spectral projector of φγ corresponding to the eigenvalue ζ ν is equal to d−1 −νj j 1 φγ . This projector is represented by a trivial cobordism with colored link j =0 ζ d γ (πν ), where the color πν is defined by πν =
d−1 1 −νj j ζ . d j =0
Using the definition of the grading, we get the following lemma. Lemma 3.1. Let V be an object in Cj ; denote by δνj the Kronecker symbol. One has the equality in Fig. 6. We denote by P T the principal SO-bundle of oriented orthonormal frames in the stabilized tangent bundle to (we could stabilize only once). We denote by γ˜ the lift in P T , using the unit tangent vector, of the embedded curve γ . Proposition 3.2. There exists a well defined action of the group H1 (P T , Z/d) on VC ( ), which maps x = [γ˜ ] to the operator ψx = (−d)γ φγ .
Spin Verlinde Formulas
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= δνj
πν
V
V
Fig. 6.
= Fig. 7. Modified band move
Proof. The Z/d module H1 (P T , Z/d) is generated by the 1-cycles γ˜ associated with embedded curves γ . A trivial circle represents the generator on the fiber; this generator has order 2. A disjoint union represents the sum. All the other relations are given by the modified band move in Fig. 7. By relation (3) the modified band move doesn’t change the number of components modulo 2, hence by relation (2) ψx is well defined by the formula ψx = (−d)γ φγ . Here γ is an embedded curve such that the lift γ˜ represents x. A crossing resolution changes by ±1 the number of components, hence in the above formula we can use an immersed curve as well. If γ , γ represent x and x , then we can isotope γ so that γ ∪ γ is an immersed curve. This shows that for all x, x , one has ψx ψx = ψx+x .
As a consequence, we have a decomposition of VC ( ) indexed by the group H 1 (P T , Z/d) identified with the characters on H1 (P T , Z/d). Recall that we have chosen a primitive d th root of unity denoted by ζ . A vector v belongs to the component indexed by σ if and only if for every x ∈ H1 (P T , Z/d) one has ψx v = ζ σ (x) v . Since the generator of the fiber acts by −1, only the classes whose restriction to the fiber is non-trivial, i.e. Spin(Z/d) structures, will correspond to non-trivial summands. If σ is a modulo d spin structure on the genus g oriented surface g , we denote by VC ( g , σ ) the corresponding summand and by dC (g, σ ) its dimension, VC ( g , σ ) = {v ∈ VC ( g ), ∀x ∈ H1 (P T , Z/d) ψx v = ζ σ (x) v}. Theorem 3.3. a) There exists a splitting of the TQFT module VC ( g ) = ⊕σ ∈Spin( g ,Z/d) VC ( g , σ ) . b) Suppose that the scalar field k has characteristic zero, then the refined Verlinde formula is the following dC ( g , σ ) = g−1
λ∈
λ 2−2g ×
g λ (aν (σ ), bν (σ )) . ( orb(λ))2
ν=1
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C. Blanchet
Here (a(σ ), b(σ )) ∈ (Z/d)g × (Z/d)g is given by the values of qσ on a sympleptic basis. If orb(λ) is even, then λ (a, b) = If orb(λ) is odd, then λ (a, b) =
1 if a and b are zero modulo |Stab(λ)|, 0 otherwise.
2a
1 |Stab(λ)| 2 (−1)
2b |Stab(λ)|
if a and b are zero modulo
0 otherwise.
|Stab(λ)| , 2
Remark. Any element in Stab(λ) has quantum dimension equal to one. In the case where d = −1, the group Stab(λ) is generated by an even power of , and orb(λ) is even. We do not know examples with d = −1. Remark. If the scalar field k has characteristic p > 0, then statement b) computes the dimension mod. p. Proof. The formula in a) follows from the decomposition of the vector space VC ( g ) described above. Moreover the dimension dC ( g , σ ) of a summand is the trace of the corresponding projector. This projector can be represented by a cobordism [0, 1] × g in which we have inserted a convenient skein element. By a standard TQFT argument we get dC ( g , σ ) = ZC (S 1 × g , skein element) . The 3-manifold S 1 × g is obtained by surgery on the borromean link with 2g + 1 components represented in Fig. 8 [20, Th. 14.12]. In this presentation, a meridian around the bigger component corresponds to S 1 × pt, and the 2g meridians around the other components correspond to a system of 2g fundamental curves in g ; these curves are framed by using the meridian disc. The skein element which arises here is represented by these 2g curves, decorated with some πν . If d = 1, then ν is the value of the quadratic form qσ on the curve, and if d = −1, then ν is the value of σ on the 1-cycle represented by the curve. By using (4) and Lemma (3.1) we get dC ( g , σ ) = D−(2g+2) λ Bλ = −1−g λ Bλ , (5) λ
λ
where Bλ is the invariant of the colored borromean link in Fig. 8. Here (a1 , b1 ), . . . , (ag , bg ) are given by the values of the quadratic form qσ on the corresponding curves if d = 1, and are equal to the value of σ on the 1-cycle represented by the curve if d = −1. Recall that the cyclic group generated by the object , identified with Z/d, acts on the set of (representatives of) isomorphism classes of simple objects. If j is in the stabilizer subgroup of λ, then we choose a basis for the 1-dimensional vector spaces H omC (j , λ∗ ⊗ λ) and the dual basis for H omC (λ∗ ⊗ λ, j ). We denote these bases by the trivalent vertices in Fig. 9. We then have the relations in Fig. 10; recall that in the case d = −1, j must be even.
Spin Verlinde Formulas
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b1 a1
bg ag λ
Fig. 8. Colored borromean link
λ
λ
j
j
λ
λ
Fig. 9. Trivalent vertices
Proposition 3.8 below is the key point in the computation. By using this proposition, we get Bλ = λ
g
ν=1 j,j ∈Stab(λ)
ζ
j aν +j bν
(−1)jj 2 . λ 2 d 2
(6)
Let l = orb(λ) and l = dl = |Stab(λ)|. The stabilizer subgroup is then Stab(λ) = {ls, 0 ≤ s < l }. If l is even then j,j ∈Stab(λ) ζ j aν +j bν is zero unless ζ laν = ζ lbν = 1, and we get j,j ∈Stab(λ)
ζ
j aν +j bν
(−1)
jj
=
|Stab(λ)|2 if aν ≡ bν ≡ 0 mod. |Stab(λ)|, 0 otherwise.
If l is odd, then we decompose the sum j,j ∈Stab(λ) ζ j aν +j bν (−1)jj according to the parity of the indices. The sum is zero if ζ 2laν = 1 or ζ 2lbν = 1. Four cases remain to consider according to ζ laν = ±1, ζ lbν = ±1.
14
C. Blanchet
λ λ
j
λ
1 = <λ>
λ
j λ
j
λ =
j
Fig. 10. Relations for trivalent vertices
Case aν ≡ bν ≡ 0 mod. |Stab(λ)|.
ζ j aν +j bν (−1)jj =
j,j ∈Stab(λ)
=
+
+
j,j even j even ,j odd l2 l2 l2 l2
−
j even ,j odd
j,j odd
+ + − 4 4 4 4 2 l . = 2
Case aν ≡ bν ≡
|Stab(λ)| 2
mod. |Stab(λ)|.
ζ j aν +j bν (−1)jj =
j,j ∈Stab(λ)
=
+
+
j,j even j even ,j odd l2 l2 l2 l2
4
=−
−
l2 2
4 .
−
4
−
4
j even ,j odd
−
j,j odd
Spin Verlinde Formulas
15
Case aν ≡ 0 , bν ≡ |Stab(λ)| mod. |Stab(λ)|. 2 ζ j aν +j bν (−1)jj = + j,j ∈Stab(λ)
+
j,j even j even ,j odd l2 l2 l2 l2
−
j even ,j odd
j,j odd
+ − + 4 4 4 4 2 l = . 2
=
Case bν ≡ 0 , aν ≡ |Stab(λ)| mod. |Stab(λ)|. 2 ζ j aν +j bν (−1)jj = + j,j ∈Stab(λ)
=
+
j,j even j even ,j odd l2 l2 l2 l2
−
j even ,j odd
j,j odd
− + + 4 4 4 4 2 l = . 2 In all cases we get the formula below. ζ j aν +j bν (−1)jj = λ (aν , bν ) |Stab(λ)|2 .
(7)
j,j ∈Stab(λ)
In the case where d = −1, then l is even, l = |Stab(λ)| divides d2 and we have λ (aν , bν ) = λ (aν + d2 , bν + d2 ). So that we may define (a1 , b1 ), . . . , (ag , bg ) by the values of the quadratic form qσ as well. We will now establish statement b), dC ( g , σ ) =
−1−g
λ
2−2g
λ
=
g−1
λ
λ
g
λ (aν , bν ) |Stab(λ)|2
ν=1
2−2g
2 d2
g λ (aν , bν ) . orb(λ)2
ν=1
Lemma 3.4. For any i ∈ Z/d the subcategory Ci contains at least one simple object, and for any simple object λi in Ci , one has λi i+j = λi ⊗ j . In a modular category the dimension of a simple object is non-zero, hence we have that for any i, 1 i = 0 = . (8) d Proof. Let ν be a generator for the subgroup of Z/d formed with all i such that Ci con tains at least one non-trivial object. Suppose that ν has order d , then d is a simple object whose contribution in the S matrix is the same as that of the trivial. This object is isomorphic to the trivial, and we deduce that d = d. This proves the first part of the lemma. The second part follows from the graded sliding property (see [19, Sect. 1.3]).
Lemma 3.5. Let λ be a simple object in C, for any i in Z/d the following morphism is non-zero if and only if λ is isomorphic to j for some j ,
16
C. Blanchet
Ωi λ Proof. If λ is equal to j then the morphism is equal to d1 ζ ij 1λ , and so is not zero. Suppose now that for some simple object λ the above morphism is not zero. By using Lemma 3.4, we obtain a scalar tλ such that
Ωi
Ω0 = tλi
λ
λ
Note that tλd = 1, hence there exists j such that tλ = ζ j . By the graded sliding property we deduce that the contribution of λ in the S matrix is the same as that of j , and we get the required isomorphism.
Lemma 3.6. Let λ be a simple object in C; then one has the relation in Fig. 11. Proof. We first use the domination axiom. The decomposition of the identity of λ∗ ⊗ λ is given by a so-called fusion formula (see e.g. [7, Sect. 1.2]). Note that in this formula the multiplicity of an invertible object is one if it belongs to the stabilizer subgroup of λ and zero otherwise. We then apply Lemma 3.5. The result follows.
Lemma 3.7. For i, j in Stab(λ), one has the relation in Fig. 12. Proof. The first equality uses the defining property of a modulo d spin modular category. The second one comes from the definition of the trivalent vertices.
Proposition 3.8. The formula in Fig. 13 holds. Proof. By using Lemma 3.6 twice (firstly for the component colored by b ), we get the formula in Fig. 14. After an isotopy, we apply Lemma 3.7. The result follows.
λ
λ
λ
a =
j ∈Stab(λ) ζ
j
aj d
λ Fig. 11.
λ
λ
Spin Verlinde Formulas
17
i
i
λ
λ
λ
j
λ
ij = (−1)2
= (−1)ij j
Fig. 12.
b a =
λ
j,j ∈Stab(λ)
2 jj ζ aj +bj (−1)2 2
λ
d
λ
Fig. 13.
j
lhs =
j,j ∈Stab(λ) ζ
aj +bj 2 d2
j
λ
Fig. 14.
4. Verlinde Formulas for Type A Modular Categories 4.1. The SU (N, K) modular category. We first consider the so-called SU (N, K) modular category. The construction can be done either from the representation theory of the quantum group Uq sl(N ) at a convenient root of unity [2, 33, 4] or from skein theory [36, 11]. In the following we will use Young diagrams to denote the corresponding simple object. Here a Young diagram (or partition) λ is a finite non-increasing sequence of
18
C. Blanchet
non-negative integers. A cell for this partition is a pair c = (i, j ) with 1 ≤ j ≤ λi . We denote by λ∨ the tranpose of λ; (i, j ) is a cell in λ∨ if and only if (j, i) is a cell in λ. The content and hook-length for a cell c = (i, j ) are defined respectively by cn(c) = j − i , hl(c) = λi + λ∨ j −i−j +1. The size of λ is |λ| = i λi . The following theorem is proved in [11]. The result can also be obtained from [4, Th. 3.3.20] (AN−1 case). Theorem 4.1. Let N, K ≥ 2. Suppose that a is a 2N (N + K)th root of unity in the scalar field and s = a −N . There exists a modular category C SU (N,K) whose set of distinguished simple objects is N,K = {λ = (λ1 , . . . , λN ), K ≥ λ1 ≥ · · · ≥ λN−1 ≥ λN = 0} . The quantum dimension and framing coefficient of a simple object λ ∈ N,K are n −s −n given by the following formulas (here [n] = ss−s −1 denotes the quantum integer). λ =
[N + cn(c)] , [hl(c)]
cells
= a |λ| s N|λ|+2 2
λ
cells cn(c)
λ
.
Remark. 1. In the quantum group approach, a Young diagram in N,K gives a highest weight module, which is irreducible and has non-zero quantum dimension. The quantum dimension follows from Weyl’s character formula and computation with symmetric functions in [21, Sect. I.3]. The value of the twist is obtained by the action of the Drinfeld quantum Casimir. 2. In the skein theoretic approach the Young diagram gives a minimal idempotent in Hecke algebra (the deformation of the Young symmetrizer in the symmetric group algebra); this idempotent becomes a simple object in the so-called Karoubi completion of the Hecke category. We denote by VN,K ( g ) the TQFT vector space, associated with a genus g surface
g , for the modular category C SU (N,K) , and by dN,K (g) its rank. We give below the well known computation for this formula. Theorem 4.2. The rank dN,K (g) is equal to the Verlinde number for the group SU (N ) at level K, dN,K (g) = VSU (N) (K, g) = g−1 (N + K)(N−1) N
λ∈N,K 1≤i<j ≤N
π 2 sin (λi − i − λj + j ) N+K
2−2g .
Spin Verlinde Formulas
19
Proof. We can use Turaev’s formula [30, Cor. 12.1.2]. Note that this formula computes the TQFT-invariant of the manifold S 1 × g , dN,K (g) =
g−1 λ 2
λ∈N,K
λ 2−2g .
λ∈N,K
The computation is achieved with the lemma below. Statement a) is a standard fact on symmetric functions; statement b) is contained e.g. in the proof of Lemma 2.8 in [11]. Note that the result does not depend on the choice of the root of unity with required order; it is also unchanged if s is replaced by s˜ = −s. The formula agrees with the Verlinde number VSU (N) (K, g) [5, 28].
Lemma 4.3. λ 2 =
a)
aρ+λ a ρ+λ aρ a ρ
with ρ = (N − 1, N − 2, . . . , 0) and, for l = (l1 , . . . , lN ), al = det s 2(i−1)lj . 1≤i,j ≤N
b)
λ∈N,K
λ 2 =
N (N + K)N−1 . aρ a ρ
4.2. Spin decomposition of the Verlinde formula for SU (N, K) modular category. In the category C SU (N,K) , the object (K) (a K cells Young diagram with only one row) is an invertible object whose order is N . It is a generator of the group of invertible objects, and has quantum dimension 1. Its framing coefficient is equal to 2
θK = a K s NK+K(K−1) = (−a N+K )K . 2
If N = j l, and (−a N+K )Kj = −1, then the category C SU (N,K) equipped with the invertible object = (K)⊗j is a modulo l spin modular category. Recall that a is a 2N (N + K)th root of unity. Let d = gcd(N, K), N = dN , K = dK . A convenient integer j exists if and only if either d is even, N is odd and the exponent of 2 in K is even (K = 22n (2m+1)), or d is odd and the exponent of 2 in N is an even positive number. We emphasize the simplest case in the theorem below. SU (N,K) Theorem 4.4. If N is even and K = K N is an odd integer, then the category C equipped with the invertible object = (K) is a modulo N spin modular category.
The following theorem is an application of 3.3.
20
C. Blanchet
Theorem 4.5. Suppose that N is even and K = splitting of the Verlinde formula
dN,K (g) =
K N
is an odd integer. a) There exists a
dN,K (g, σ ) .
σ ∈Spin( g ,Z/d)
b) The refined Verlinde formula is the following g−1 dN,K (g, σ ) = (N + K)N−1 N
×
1≤i<j ≤N
g λ (aν (σ ), bν (σ )) ( orb(λ))2 λ∈N,K ν=1 2−2g π 2 sin (λi − i − λj + j ) . N +K
Here (a(σ ), b(σ )) ∈ Z/N )g × (Z/N )g are given by the values of qσ on a sympleptic basis. We consider the action of Z/N on the set N,K = {λ = (λ1 , . . . , λN ), K ≥ λ1 ≥ · · · ≥ λN−1 ≥ λN = 0} , given for the generator of the cyclic group Z/N by (λ1 , . . . , λN−1 , 0) −→ (K, λ1 , . . . , λN−1 ) − (λN−1 , . . . , λN−1 ) . We denote by orb(λ) the cardinality of the orbit of λ, and by Stab(λ) the stabilizer subgroup. The numbers λ (a, b) ∈ {0, 1, − 21 , 21 } are defined as follows. If orb(λ) is even, then λ (a, b) =
1 if a and b are zero modulo |Stab(λ)|, 0 otherwise.
If orb(λ) is odd, then λ (a, b) =
2a
1 |Stab(λ)| 2 (−1)
2b |Stab(λ)|
if a and b are zero modulo
0 otherwise.
|Stab(λ)| , 2
Remark. In the general case, one can use the reduction formula [11, Theorem 3.6] in order to establish a tensor product decomposition of the SU (N, K) TQFT functor VN,K , U (1)
VN,K = VN
⊗ VN,K ,
where VN,K is the TQFT functor associated with the modular category C P U (N,K) disU (1) cussed below, and VN (known as a U (1) theory) is associated with a modular category based on linking numbers. The latter involves a root of unity η whose order is 2N (resp. N ) if N is even (resp. odd); when N is even with even exponent of 2, then one can find j such that j 2 ≡ N (mod 2N ), and the category is modulo j spin modular.
Spin Verlinde Formulas
21
4.3. The P U (N, K) modular category. The so-called projective P SU (N, K) modular category was obtained for N and K coprime by restricting to simple objects in the root lattice [22, 18, 19]. The modular category C P U (N,K) (denoted by HN,K in [11]) is a generalization to the case where N and K are not required to be coprime. Let N, K ≥ 2. We suppose that in the scalar field s has order 2(N + K) if N + K is even, s has order N + K if N + K is odd. Then the Hecke category completed with idempotents and quotiented with negligible, which we denote by H N,K is semisimple. In addition to simple objects λ ∈ N,K there is an invertible simple object 1N and its tensor powers. The group of invertible objects is generated by 1N and (K) with the relation (1N )⊗K ≈ (K)⊗N . In order to apply the modularization procedure, we have to know which are the transparent simple objects [14]. This depends on the order α of (a N s)2 and the order β of (a K s −1 )2 . The set of isomorphism classes of transparent simple objects is then the group generated by (1N )⊗α and (K)⊗β . We choose the framing parameter a in such a way that this group of transparent objects is as big as possible, and that the modularization criterion is satisfied. Theorem 4.6. Set d = gcd(N, K), N = dN , K = dK , d = αβ with gcd(α, K ) = gcd(β, N ) = gcd(α, β) = 1. Suppose that a satisfies the relations (a N s)α = (−1)N+K+1 (a K s −1 )β = (−1)(N+K+1)β (such an a exists up to extension of the scalar field). There exists a modular category C P U (N,K) in which isomorphism classes of simple objects corresponds bijectively with cosets in the quotient of ˙ N,K = {(1N )⊗j ⊗ λ, 0 ≤ j < α, λ ∈ N,K } under a free action of the cyclic group of order N/α. The action of the generator is given by tensor product with (K)⊗β in the completed Homfly category H . One has to iterate β times the rule
(1N )⊗j ⊗ λ → (1N )⊗j + (K − λN−1 , λ1 − λN−1 , . . . , λN−2 − λN−1 , 0), where j ≡ j + λN−1 mod α. The quantum dimension and framing coefficient of a simple object V = (j ⊗N ) ⊗ λ are given by the following formulas: V = λ =
[N + cn(c)] , [hl(c)]
cells
V
= (a N s)Ni
2 +2|λ|
a |λ| s N|λ|+2 2
cells cn(c)
V.
We denote by VN,K ( g ) the TQFT vector space, associated with a genus g surface
g , for the modular category C P U (N,K) and by d˜N,K (g) its rank.
22
C. Blanchet
Theorem 4.7. The rank d˜N,K (g) is dN,K (g) d˜N,K (g) = . N g Proof. By Turaev’s formula we have the following. d˜N,K (g) =
g−1
V 2
V ∈ N,K
V 2−2g .
V ∈ N,K
Here N,K ⊂ ˙ N,K is a representative set of the orbits in ˙ N,K under the order N/α free cyclic action. Note that this action preserves the dimension. We get 1 d˜N,K (g) = αN
g−1
1 αN
V 2
V ∈˙ N,K
V 2−2g .
V ∈˙ N,K
Write V = (1N )⊗i ⊗ λ, 0 ≤ i < α and λ ∈ N,K , 1 d˜N,K (g) = N
V ∈N,K
g−1 V 2
1 N
V 2−2g =
V ∈N,K
dN,K (g) . N g
The following is an integral version of a reciprocity formula in [26]. Theorem 4.8. (Level-rank duality) One has d˜N,K (g) = d˜K,N (g). Proof. In the construction arising from Homfly skein theory, the parameters N and K play the same role, so that we can interchange rows and columns in the description of isomorphism classes in the modular category C P U (N,K) . We will get the same combinatorics as for the modular category C P U (K,N) . The result follows.
4.4. Spin decomposition of the Verlinde formula for C P U (N,K) . Here we consider the modular category C P U (N,K) in the spin case. This means that d = gcd(N, K) is even, and that N = Nd and K = Kd are both odd. We fix the framing parameter a as we did above. Theorem 4.9. Under the above hypothesis, the category C P U (N,K) equipped with = (K) ⊗ (1N ) is a modulo d spin modular category. Proof. In the modular category C P U (N,K) the object 1N and (K) are invertible with respective orders the coprime integers α and β. It follows that is invertible with order αβ = d. Figure 15 below shows that the twist coefficient for is the product of the two twist coefficients for 1N and (K) and a braiding coefficient between 1N and (K). Using [11, Prop. 1.11] we see that the 3 coefficients are respectively (a N s)N = (−1)β , (AK s −1 )K = (−1)α , (a N s)2NK = 1. The product is −1.
Spin Verlinde Formulas
23
=
V ⊗W
V
W
Fig. 15. Framing coefficient for a tensor product
If σ is a modulo d spin structure on the genus g oriented surface g , we denote by ˜ σ ) its dimension. By applying 3.3, we V( g , σ ) the corresponding summand and d(g, get Theorem 4.10. a) There exists a splitting of the Verlinde formula d˜N,K (g, σ ) . d˜N,K (g) = σ ∈Spin( g ,Z/d)
b) The refined Verlinde formula is the following: g−1 d˜N,K (g, σ ) = (N + K)N−1 d
×
g V (aν (σ ), bν (σ )) ( orb(V ))2
V =(1N )ι ⊗λ∈˜ N,K ν=1
2 sin (λi − i − λj + j )
1≤i<j ≤N
π N +K
2−2g .
Here V and orb(V ) are defined as before, ˜ N,K is a representative set of the quotient of ˙ N,K = {(j ⊗N ) ⊗ λ, j ∈ Z/α, λ ∈ N,K } under a free action of Z/αN . The formula in b) can be expressed as follows: g−1 1 d˜N,K (g, σ ) = (N + K)N−1 d αN 2 g αN × V (aν (σ ), bν (σ )) Orb(V ) V =(1N )ι ⊗λ∈˙ N,K ν=1 2−2g π 2 sin (λi − i − λj + j ) × . N +K 1≤i<j ≤N
We consider now the orbit Orb(V ) under the action of the group Z/α × Z/N on ˙ N,K , where (1, 0) acts by (1N )⊗ι ⊗ λ → (1N )⊗(ι+1) ⊗ λ,
24
C. Blanchet
and (0, 1) acts by (1N )⊗ι ⊗ λ → (1N )⊗(ι+λn−1 ) ⊗ ((K, λ) − λN N−1 ) . If Orb(V )/αN is even, then 1 if a and b are zero modulo |Stab(V )|, V (a, b) = 0 otherwise. If Orb(V )/αN is odd, then 2b 2a 1 |Stab(V )| |Stab(V )| if a and b are zero modulo V (a, b) = 2 (−1) 0 otherwise.
|Stab(V )| , 2
5. Cohomological Decomposition In this section we will establish the decomposition in the cohomological case. Let d be an integer, and (C, ) be a modulo d cohomological modular category. This means that the object has order d and twist coefficient θ = 1. We deduce that the quantum dimension of is d = ±1, and
= d
= d
(9)
(10)
After fixing a d th root of unity ζ , the category is Z/d graded. The Kirby color decomposes according to this grading, λ λ = j . = λ∈
j ∈Z/d
Using this grading we obtain the theorem below [11, 19]. Theorem 5.1. Let C be a modulo d cohomological modular category, and = j ∈Z/d j be the graded decomposition of the Kirby element. Provided c = (c1 , . . . , cm ) ∈ Zm is in the kernel of BL ⊗ Z/d the formula τCcoho (M, σ ) =
L(c1 , . . . , cm ) U1 () b+ U−1 () b−
is an invariant of the surgered manifold M = S3 (L) equipped with the modulo d cohomology class σ corresponding to c. Moreover, τCcoho (M, σ ). ∀M τC (M) = σ ∈Spin(M;Z/d)
Spin Verlinde Formulas
25
Following Sect. 2 we get the proposition below. Note that here the action given by a trivial curve γ colored with dφγ is trivial. Proposition 5.2. There exists a well defined action of the group H1 ( , Z/d) on VC ( ), which maps x = [γ ] to the operator ψx = (d)γ φγ . Using this action we get the decomposition theorem below. Theorem 5.3. Let (C, ) be a modulo d cohomological modular category. a) There exists a splitting of the Verlinde formula
dim(VC ( g )) =
dim(VC ( g , σ )) .
σ ∈H 1 ( g ,Z/d)
b) The refined Verlinde formula is the following dim(VC ( g , σ )) = g−1
λ 2−2g ×
λ∈
g λ (aν (σ ), bν (σ )) . ( orb(λ))2
ν=1
Here (a(σ ), b(σ )) ∈ (Z/d)g × (Z/d)g is given by the values of σ on a sympleptic basis, and 1 if a and b are zero modulo |Stab(λ)|, λ (a, b) = 0 otherwise. Proof. The decomposition a) follows from the action given in Proposition 5.2. For σ ∈ H 1 ( g , Z/d), we have dim(VC ( g , σ )) = −1−g
λ Bλ ,
λ
where Bλ is the invariant of the colored link in Fig. 8. Here (a1 , b1 ), . . . , (ag , bg ) are given by the values of σ on the corresponding curves if d = 1, and this value plus d2 if d = −1 (in this case d has to be even). The computation is done as in Sect. 3. We have Bλ = λ
g
ν=1 j,j ∈Stab(λ)
The formula follows.
ζ j aν +j bν 2 . λ 2 d2
(11)
is even, then the category Let d = gcd(N, K). If d is odd, or if d is even but NK d2 is a modulo d cohomological modular category. 2 If N = j l, and (−a N+K )Kj = 1, then the category C SU (N,K) equipped with the ⊗j invertible object = (K) is a modulo l cohomological modular category. In particuSU (N,K) is a modulo lar, if N divides K, and N is odd or K N is even, then the category C N cohomological modular category. C P U (N,K)
26
C. Blanchet
6. Some Computations We give below some computations obtained with MuPAD [25]. Our program implements the Verlinde formulas for the categories C SU (N,K) . The cardinality of the alcove increases rapidly, and we obtain results only for small values of N ,K. The function Verlinde(N, K, g) gives dN,K (g), and Spin Verl(N, K, [. . . ]) computes dN,K (g, σ ), where the value of qσ on the standard basis is the list [. . . ]. We know [13] that d2,2 (g, σ ) is 0 or 1 according to the Arf invariant of the spin structure. It would be interesting to understand the combinatorics of the formula dN,K (g, σ ) in the general case. Verlinde(2,2,1); 3 Verlinde(2,2,2); 10 Spin_Verl(2,2,[[0,0]]); 1 Spin_Verl(2,2,[[1,1]]); 0 Spin_Verl(2,2,[[0,0],[1,1]]); 0 Spin_Verl(2,2,[[1,1],[1,1]]); 1 Verlinde(2,6,1); 7 Verlinde(2,6,2); 84 Spin_Verl(2,6,[[0,0]]); 2 Spin_Verl(2,6,[[1,1]]); 1 Spin_Verl(2,6,[[0,0],[1,1]]); 4 Spin_Verl(2,6,[[0,0],[0,0]]); 6 Verlinde(4,4,1); 35 Verlinde(4,4,2); 4680 Spin_Verl(4,4,[[0,0]]); 3 Spin_Verl(4,4,[[1,0]]); 2 Spin_Verl(4,4,[[1,1]]); 2 Spin_Verl(4,4,[[2,2]]); 2 Spin_Verl(4,4,[[0,0],[0,0]]); 24 Spin_Verl(4,4,[[1,0],[0,0]]); 18
Spin Verlinde Formulas
27
Spin_Verl(4,4,[[1,0],[1,0]]); 18 Spin_Verl(4,4,[[2,2],[0,0]]); 20 Verlinde(6,6,1); 462 Verlinde(6,6,2); 30660988 Spin_Verl(6,6,[[0,0]]); 14 Spin_Verl(6,6,[[1,0]]); 13 Spin_Verl(6,6,[[2,0]]); 13 Spin_Verl(6,6,[[1,1]]); 12 Spin_Verl(6,6,[[2,2]]); 13 Spin_Verl(6,6,[[3,0]]); 14 Spin_Verl(6,6,[[3,3]]); 12 Spin_Verl(6,6,[[0,0],[0,0]]); 23718 Spin_Verl(6,6,[[1,0],[0,0]]); 23678 Spin_Verl(6,6,[[1,0],[0,0]]); 23624 Spin_Verl(6,6,[[2,0],[0,0]]); 23678 Spin_Verl(6,6,[[2,2],[0,0]]); 23678 Spin_Verl(6,6,[[3,0],[0,0]]); 23718 Spin_Verl(6,6,[[3,3],[0,0]]); 23648 References 1. Andersen, J., Masbaum, G.: Involutions on moduli spaces and refinements of the Verlinde formula. Math. Ann. 314(2), 291–326 (1999) 2. Andersen, H., Paradowski, J.: Fusion category arising from semisimple Lie algebras. Commun. Math. Phys. 169(3), 563–588 (1995) 3. Atiyah, M. F.: Riemann surfaces and spin structures. Ann. Ecole Norm. Sup. (4)4, 47–62 (1971) 4. Bakalov, B., Kirillov, A.: Lecture on tensor categories and modular functors. Univ. Lecture Series No.21, Providence, RI: AMS 2001 5. Beauville, A.: Conformal blocks, fusion rules and the Verlinde formula. Israel Math. Conf. Proceedings, Vol.9, 75–96 (1996) 6. Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions. Commun. Math. Phys. 164, 385–419 (1994) 7. Beliakova, A., Blanchet, C.: Modular categories of types B,C and D. Comment. Math. Helv. 76, 467–500 (2001)
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8. Bismut, J-M., Labourie, F.: Formules de Verlinde pour les groupes simplement connexes et g´eom´etrie sympleptique. CRAS, t. 325, S´erie I, 1009–1014 (1997) 9. Bismut, J-M., Labourie, F.: Sympleptic geometry and the Verlinde formulas. In: Surveys in differential geometry: differential geometry inspired by string theory, Boston, MA: Int. Press, 1999, pp. 97–311 10. Blanchet, C.: Refined quantum invariants for three-manifolds with structure. In: Knot Theory, Banach Center Pub. Vol. 42, Warsaw: Polish Acad. of Sci, 11–22 (1998) 11. Blanchet, C.: Hecke algebras, modular categories and 3-manifolds quantum invariants. Topology, 39, 193–223 (2000) 12. Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Topological Quantum Field Theories derived from the Kauffman bracket. Topology 34(4), 883–927 (1995) 13. Blanchet, C., Masbaum, G.: Topological quantum field theories for surfaces with spin structure. Duke Math. J 82, 229–267 (1996) 14. Brugui`eres, A.: Cat´egories pr´emodulaires, modularisations et invariants des vari´et´es de dimension 3. Math. Ann. 316(2), 215–236 (2000) 15. Faltings, G.: A proof of the Verlinde formula. J. Alg. Geometry 3, 347–374 (1994) 16. Johnson, D.: Spin structures and quadratic forms on surfaces. J. London Math Soc. (2) 22, 365–377 (1980) 17. Kohno, T., Takata, T.: Level-Rank Duality of Witten 3-manifolds invariants. Adv. Studies in Pure Math. 24, Progress in Algebraic Combinatorics, Orlando, FL: Acad.Press, 1996 pp. 243–264 18. Le, T.: Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion. http://arxiv.org/list/math.QA/0004099, 2000 19. Le, T., Turaev, V.: Quantum groups and ribbon G-categories. J. Pure Appl. Algebra 178 (2), 169–185 (2003) 20. Lickorish, W.B.R.: An Introduction to Knot Theory. Grad. Texts in Math. 175, Berlin-HeidelbergNew York: Springer Verlag, 1997 21. Macdonald, I. G.: Symmetric functions and Hall polynomial. 2nd ed. , Oxford: Oxford Science Pub 1995 22. Masbaum, G., Wenzl, H.: Integral modular categories and integrality of quantum invariants at roots of unity of prime order. J. Reine Angew. Math. 505, 209–235 (1998) 23. Milnor, J.: Spin structures on manifolds. L’Enseignement Math. 9, 198–203 (1963) 24. M¨uger, M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151–201 (2000) 25. MuPAD: The Open Computer Algebra System. Sciface Software, www.mupad.de. 26. Oxbury, W. M., Wilson, S. M. J.: Reciprocity laws in the Verlinde formulae for the classical groups. Trans. AMS 348(7), 2689–2710 (1996) 27. Kassel, C., Rosso, M., Turaev, V.: Quantum groups and knots invariants. Panoramas et Synth`eses No 5, Paris: Soc. Math. France, 1997 28. Sorger, C.: La formule de Verlinde. S´eminaire Bourbaki 794, 1994 29. Sawin, S.: Quantum groups at roots of unity and modularity. http://arxiv.org/list/math.QA/0308281, 2003 30. Turaev, V.: Quantum invariants of knots and 3-manifolds. De Gruyter Studies in Math. 18, Berlin: De Gruyler, 1994 31. Turaev, V.: Homotopy field theory in dimension 2 and crossed groups-algebras. http:// arxiv.org/list/math.QA/9910010, 1999 32. Turaev, V.: Homotopy field theory in dimension 3 and crossed groups-categories. http:// arxiv.org/list/math.GT/0005291, 2000 33. Turaev, V., Wenzl, H.: Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Int. J. of Math. 4(2), 323–358 (1993) 34. Turaev, V., Wenzl, H.: Semisimple and modular categories from link invariants. Math. Ann. 309, 411–461 (1997) 35. Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B 300(3), 360–376 (1988) 36. Yokota, Y.: Skeins and quantum SU (N) invariants of 3-manifolds. Math. Ann. 307, 109–138 (1997) Communicated by Y. Kawahigashi
Commun. Math. Phys. 257, 29–42 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1345-2
Communications in
Mathematical Physics
Initial Data Engineering Piotr T. Chru´sciel1, , James Isenberg2, , Daniel Pollack3, 1
Dept. de Mathématiques, Université de Tours, 37041 Tours Codex 1, France. E-mail:
[email protected] 2 Department of Physics, University of Oregon, Eugene, OR 97403, USA. E-mail:
[email protected] 3 Mathematics Department, University of Washington, Box 354350, Seattle, WA 98195-4350, USA. E-mail:
[email protected] Received: 27 January 2004 / Accepted: 3 February 2005 Published online: 15 April 2005 – © Springer-Verlag 2005
Abstract: We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface. 1. Introduction Let (Ma , γa , Ka ), a = 1, 2, be two (arbitrary dimensional) general relativistic initial data sets; by this we mean that each γa is a Riemannian metric on the n dimensional manifold Ma , while each Ka is a symmetric two-covariant tensor field on Ma . Such a data set is called vacuum data if it satisfies the vacuum Einstein constraint equations R(γ ) − (2 + |K|2γ − (trγ K)2 ) = 0,
(1.1)
Di (K − trγ Kγ ) = 0,
(1.2)
ij
ij
where R(γ ) is the scalar curvature (Ricci scalar) of the metric γ , and is the cosmological constant. The vacuum local gluing problem can be formulated as follows: Let pa ∈ Ma be two points, and let M be the manifold obtained by removing from Ma geodesic balls of radius around the pa , and gluing in a neck Sn−1 × I , where I is an interval. Can one find vacuum initial data (γ , K) on M which coincide with the original vacuum data away from a small neighborhood of the neck?
Partially supported by a Polish Research Committee grant 2 P03B 073 24 Partially supported by the NSF under Grants PHY-0099373 and PHY-0354659 Partially supported by the NSF under Grant DMS-0305048 and the UW Royalty Research Fund
30
P. T. Chru´sciel, J. Isenberg, D. Pollack
It is natural to pose the same question in field theoretical models with matter. A formulation that avoids the issue of specifying the precise nature of the matter fields is obtained if we represent these fields in the initial data set by the matter energy density function ρ and the matter energy-momentum vector J , requiring that they satisfy the dominant energy condition1 ρ ≥ |J | .
(1.3)
The Einstein-matter constraints then relate ρ and J to the gravitational fields via the following: 16πρ = R(γ ) − (2 + |K|2γ − (trγ K)2 ) ,
(1.4)
16πJ j = 2Di (K ij − trγ Kγ ij ) .
(1.5)
As a variation of the local gluing problem, one has the wormhole creation problem, or the ˜ γ˜ , K), ˜ vacuum wormhole creation problem: one starts with a single initial data set (M, ˜ As before one forms the new manifold M by and chooses a pair of points pa ∈ M. replacing small geodesic balls around these points by a neck S n−1 × I , and one asks for the existence of initial data on M which satisfy either the vacuum or the Einstein-matter constraints, and which coincide with the original data away from the neck region. It is easily seen that for certain special sets of initial data, such constructions are not possible: consider, for example, the flat initial data set (R3 , δ, 0) associated with Minkowski space-time. It follows from the positive energy theorem that this set of data cannot be glued to any data on a compact manifold without globally perturbing the metric (so that the mass is nonzero). The object of this work is to show that the above gluing constructions can be performed for generic initial data sets. To make our notion of genericity precise, some terminology is needed. Let P denote the linearisation of the map which takes a set of data (g, K) to the constraint functions appearing in (1.1)–(1.2), and let P ∗ be its formal adjoint. Recall that a Killing Initial Data (KID) is defined as a solution (N, Y ) of the set of equations P ∗ (N, Y ) = 0. These equations are given explicitly by 2(∇(i Yj ) − ∇ l Yl gij − Kij N + trK Ngij ) l l ∇ Y + K q ∇ Y l g − Ng + ∇ ∇ N Y K − 2K ∇ 0= (1.6) l ij l q ij ij i j l (i j ) . +(∇ p Klp gij − ∇l Kij )Y l − NRic (g)ij +2N K l i Kj l − 2N (tr K)Kij We shall denote by K ( ) the set of KIDs defined on an open set (note that we impose no boundary conditions on (N, Y ).) In a vacuum space-time (M , g) (possibly with non-zero cosmological constant) the KIDs on a spacelike hypersurface are in one-to-one correspondence with the Killing vectors of g on the domain of dependence of [21]. A similar statement, with an appropriately modified equation for the KIDs, holds in electro-vacuum for appropriately invariant initial data for the gravitational and electromagnetic fields. (The reader is referred to [6] for comments about such data for general matter fields.) 1 Recall that (1.3) might fail when quantum phenomena are taken into account. We note that the local gluing problem is trivial if no energy restrictions, or matter content restrictions, are imposed, as then both the metric and the extrinsic curvature can be glued together in many different ways.
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We note that the gluing problem is in fact a special case of the wormhole creation problem if one allows M˜ to be a non-connected manifold. Hence from now on we shall ˜ a = 1, 2. The first main assume that M˜ has either one or two components, with pa ∈ M, result of our paper concerns vacuum initial data: ˜ γ˜ , K) ˜ be a smooth vacuum initial data set, and consider two Theorem 1.1. Let (M, ˜ open sets a ⊂ M with compact closure and smooth boundary such that the set of KIDs, K ( a ), is trivial. Then for all pa ∈ a , > 0 and k ∈ N there exists a smooth vacuum initial data ˜ in a C k × C k set (M, γ (), K()) on M such that (γ (), K()) is -close to (γ˜ , K) ˜ topology away from B(p1 , ) ∪ B(p2 , ). Moreover (γ (), K()) coincides with (γ˜ , K) away from 1 ∪ 2 . The hypothesis of smoothness has been made for simplicity. Similar results, with perhaps some finite loss in differentiability, can be obtained for initial data sets with finite Hölder or Sobolev differentiability. Some comments about the no-local-KIDs condition K ( a ) = {0} are in order. As noted above, this is equivalent to the condition that there are no Killing vectors defined on the domain of dependence of the regions a in the associated vacuum space-time. First, the result is sharp in the following sense: as discussed above, initial data for Minkowski space-time cannot locally be glued to anything which is non-singular and vacuum. This meshes with the fact that for Minkowskian initial data, we have K ( ) = {0} for any open set . Next, it is intuitively clear that for generic space-times there will be no locally defined Killing vectors, and several precise statements to this effect have been proved in [7]. Thus, our result can be interpreted as the statement that for generic vacuum initial data sets the local gluing can be performed around arbitrarily chosen points pa . In particular it follows from the results in [7] that the collection of initial data with generic regions a satisfying the hypotheses of Theorem 1.1 is not empty. Further, it follows from the results here together with those in [18] and in [7] that the following initial data sets can always be glued together, near arbitrary points, after a (perhaps global) perturbation which is -small away from the gluing region: • initial data containing an asymptotically flat region • initial data containing a conformally compactifiable CMC region • CMC initial data on a compact boundaryless manifold. Let us denote by N ( ) the set of functions N satisfying, on , the second of equations (1.6) with K ≡ 0. Theorem 1.1 has the following purely Riemannian “timesymmetric” counterpart: ˜ γ˜ ) be a smooth Riemannian manifold with non-positive constant Theorem 1.2. Let (M, scalar curvature ν, and consider two open sets a ⊂ M˜ with compact closure and smooth boundary such that N ( a ) = {0} . Then for all pa ∈ a , > 0 and k ∈ N there exists a Riemannian manifold (M, γ ()) with scalar curvature ν such that γ () is -close to γ˜ in a C k topology away from B(p1 , ) ∪ B(p2 , ). Moreover γ () coincides with γ˜ away from 1 ∪ 2 .
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The proof is a simplified version of that of Theorem 1.1; we leave the details to the reader. This result is the local counterpart of the gluing theorem of Joyce [20] for ν < 0, and appears to be completely new in the case ν = 0. A noteworthy consequence of Theorem 1.1, proved in Sect. 5.1, is the following: Corollary 1.3. There exist vacuum maximal globally hyperbolic space-times with compact Cauchy surfaces which contain no compact boundaryless spacelike hypersurfaces with constant mean curvature. It is clear that there exist equivalents of Theorem 1.1 in non-vacuum field theoretical models. However, proofs for such models require a case-by-case analysis of the corresponding gluing and KID equations. It is therefore noteworthy that one can make a general statement assuming only the dominant energy condition, which we use in its (n + 1)-dimensional formulation: Tµν X µ Y ν ≥ 0 for all timelike future directed vectors X µ and Y µ .
(1.7)
The second main result of this paper reads: Theorem 1.4. Consider a smooth solution (M , g) of the Einstein field equations Gµν = 8π Tµν , with one or two connected components, and with matter fields satisfying the dominant energy condition (1.7). Let M˜ be a spacelike hypersurface in M with induced data ˜ and let (γ˜ , K), ˜ a = 1, 2, be two points at which the inequality (1.7) is strict. pa ∈ M, Then for all > 0 there exists a smooth initial data set (M, γ (), K()) on M satisfying ˜ away from the dominant energy condition such that (γ (), K()) coincides with (γ˜ , K) B(p1 , ) ∪ B(p2 , ). The reader will have observed that Theorem 1.1 concerns initial data sets only, while in Theorem 1.4 the starting point is a space-time. This is related to the fact that we have ˜ g, ˜ not made any assumptions on the matter fields except energy dominance. If (M, ˜ K) has constant mean curvature, then the proof of Theorem 1.4 is such that we could restate the result purely in terms of initial data, with no reference to the space-time (M , g). Theorems 1.1 and 1.4 are established in Sect. 4. The proofs are a mixture of gluing techniques developed in [17–19] and those of [12–14]. In fact, the proof proceeds via a generalisation of the analysis in [18, 19] to compact manifolds with boundary; this is carried through in Sect. 2 in vacuum with cosmological constant = 0, and in Sect. 3 with matter and ∈ R. These results may be of independent interest. In order to have CMC initial data near the gluing points, which the analysis based on [18] requires, we make use of the work of Bartnik [2] on the plateau problem for prescribed mean curvature spacelike hypersurfaces in a Lorentzian manifold. 2. The (Global) Gluing Construction for the Vacuum Constraints with Λ = 0 for Manifolds with Boundaries In this section we formulate some generalisations of the results in [18] and [19] to vacuum initial data sets on manifolds with boundary, with vanishing cosmological constant; the vacuum case with = 0 will be covered by the analysis in Sect. 3. Although the paper [18] only treats the case n = 3, since the generalization to higher dimensions is
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not difficult (the necessary modifications are discussed in [17]), we work here in general ˜ γ˜ , K), ˜ where M˜ has non-empty dimension n ≥ 3. We begin with an initial data set (M, smooth boundary ∂ M˜ and we assume first that K˜ has constant trace τ˜ = trγ˜ K˜ (i.e., these are constant mean curvature, or CMC, initial data sets). Decomposing K˜ into its trace and trace-free components, we write K˜ = µ˜ + τn˜ γ˜ . Since trγ˜ K˜ is constant, the vacuum momentum constraint equation implies that µ˜ is divergence free as well (i.e., µ˜ is a transverse–traceless tensor). We “mark” M˜ with two points pa , a = 1, 2, about which we will perform the gluing. The global gluing construction can be carried out in this setting, with Dirichlet boundary conditions on the perturbation terms which arise in applying the conformal method (see (2.1) and (2.2) below), generalizing the result of [18]. ˜ γ˜ , K; ˜ pa ) be a smooth, marked, constant mean curvature soluTheorem 2.1. Let (M, tion of the Einstein vacuum constraint equations with cosmological constant = 0 ˜ an n-manifold with boundary. Then there is a geometrically natural choice on M, of a parameter T and, for T sufficiently large, a one-parameter family of solutions (MT , T , KT ) of the Einstein constraint equations with the following properties. The n-manifold MT is constructed from M˜ by adding a neck connecting the two points p1 and p2 . For large values of T , the Cauchy data ( T , KT ) is a small perturbation of the initial ˜ away from small balls about the points pa . In fact, for any > 0 and Cauchy data (γ˜ , K) ˜ as T → ∞ in C k M \ (B(p1 , ) ∪ B(p2 , )) . 2 k ∈ N we have ( T , KT ) → (γ˜ , K) Proof: These solutions are constructed via the conformal method following the technique developed in [18]. The adaptation of the proof of Theorem 1 of [18] to allow for initial data on manifolds with boundary requires only minor variations which we indicate here. The construction begins with a conformal deformation of the initial data within small balls about the points pa , a = 1, 2. The metric is conformally deformed to make deleted neighborhoods of these points asymptotically cylindrical. One then truncates these neighborhoods at a distance T (in the asymptotically cylindrical metric, for T large) and identifies the remaining ends to form the new manifold MT with metric γT . The first variation in the proof occurs when deforming the approximate transverse– traceless µT formed by gluing the conformally transformed µ˜ across the neck via cut-off functions. This requires solving (with appropriate estimates) the elliptic system LX = W, where W = div γT µT is supported near the center of the asymptotic cylinder, L = −div γT ◦ D and DX = 21 LX γT − n1 (div γT X)γT is the conformal Killing operator applied to the (unknown) vector field X. In [18] the required uniform invertibility of L is established under a nondegeneracy condition which amounts to the absence of conformal Killing vectors fields (which are in the kernel of L) vanishing at pa . When M˜ has a non-empty boundary we are actually interested in solutions to the boundary value problem LX = W in MT (2.1) X = 0 on ∂MT . 2 One should note the absence of any nondegeneracy condition in Theorem 2.1. As is evident in the proof, this is accounted for by the imposition of Dirichlet boundary conditions.
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The core to solvability of this problem is provided by Theorem 2 of [18]. The proof in the present setting is identical to the one there with the exception that the step where the nondegeneracy condition (Definition 1 of [18]) is evoked is now replaced by the nonexistence of conformal Killing fields which vanish on the boundary, ∂ M˜ (see, e.g., Prop. 6.2.2 of [1]). The required estimates on the solution follow from Corollary 1 of [18] coupled with the boundary Schauder estimates. Setting σT = DX and µ˜ T = µT − σT , we see that µ˜ T is our desired transverse-traceless tensor. The other modification occurs in solving the nonlinear boundary value problem NT (ψT + ηT ) = 0 in MT , (2.2) ηT = 0 on ∂MT , where ηT is presumed to be a small perturbation of an explicit approximate solution ψT , and NT is the Lichnerowicz operator, NT (ψ) = T ψ −
−3n+2 n−2 n−2 n − 2 2 n+2 RT ψ + |µ˜ T |2 ψ n−2 − τ ψ n−2 . (2.3) 4(n − 1) 4(n − 1) 4n
Equation (2.2) is solved by means of a contraction mapping argument. The key ingredient is a good understanding of the linearised operator LT on MT . LT is the operator n−2 3n − 2 − 4(n−1) R(γT )+ |µ˜ T |2 ψT n−2 LT = γT − 4(n − 1) n−2 4 (n − 1)(n + 2) 2 n−2 . (2.4) τ ψT + n(n − 2) The basic point is to show that, corresponding to the solutions to the boundary value problem LT η = f in MT , η = 0 on ∂MT , we have an isomorphism between certain weighted Hölder spaces on MT where the weight factor controls decay/growth across the neck, and moreover for a certain range of weights, there is a T0 such that this map has a uniformly bounded inverse for all T ≥ T0 . The proof of this follows §5 of [18] and relies on the fact that the boundary value problem
˜ γ η − |µ|2γ + n1 τ 2 η = 0 in M, (2.5) ˜ η = 0 on ∂ M, has no non-trivial solutions. The linear operator appearing in (2.5) is precisely the lin˜ γ˜ , K). ˜ earised Lichnerowicz operator about the original solution (M, Letting ψ˜ T = ψT + ηT be the solution to (2.2) one finds that the desired solution to the constraint equations is then given by 4
T = ψ˜ Tn−2 γT
and
KT = ψ˜ T−2 µ˜ T +
4 1 τ ψ˜ Tn−2 γT . n
The fact that these solutions converge uniformly to the original initial data sets in ˜ away from small balls about the points p1 , p2 follows from the calculations C k,α (M) of §8 of [18] together with the boundary Schauder estimates.
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The gluing construction of [19], which only requires the initial data to have constant mean curvature in small balls about the points at which the gluing is to be done, also easily generalizes to manifolds with boundary. To show this, we need to introduce the notion of nondegeneracy for solutions of the constraint equations which are not necessarily CMC on manifolds with boundary. We do this in the context of the conformal method for non-CMC data, which works as follows: Given a fixed background metric γ , a trace-free symmetric tensor µ, and a function τ , if we can solve the coupled equations 2 −3n+2 n−2 n−2 2 n+2 n−2 = 0, γ φ − 4(n−1) Rγ φ + 4(n−1) µ + DW φ n−2 − n−2 4n τ φ LW − (div µ −
2n n−1 n−2 ∇τ ) n φ
= 0,
for a positive function φ and a vector field W , then the initial data 4
γ˜ = φ n−2 γ ,
4 = φ −2 (µ + DW ) + τ φ n−2 γ , K n
satisfies the ( = 0) vacuum Einstein constraints (1.1)–(1.2). The first of these is again referred to as the Lichnerowicz equation. We write this coupled system as N (φ, W ; τ ) = 0. The mean curvature τ is emphasized here, while the dependence of N on γ and µ is suppressed. We are interested here in the boundary value problem ˜ N (φ, W ; τ ) = 0 in M, ˜ (2.6) φ = 1 on ∂ M, ˜ W = 0 on ∂ M. The linearization L of N in the directions (φ, W ) (but not τ ) is of central concern. We consider this linearization relative to a specified choice of Banach spaces X and Y, each consisting of scalar functions and vectors fields vanishing on the boundary. If our manifold were not compact then one would also build into X and Y appropriate asymptotic conditions. Definition 1. A solution to the constraint equation boundary value problem (2.6), is nondegenerate with respect to the Banach spaces X and Y provided L : X → Y is an isomorphism. The main result of the first gluing paper [18] shows that any two nondegenerate solutions of the vacuum constraint equations with the same constant mean curvature τ can be glued. For compact CMC solutions on manifolds without boundary, nondegeneracy is equivalent to K ≡ 0 together with the absence of conformal Killing fields, while asymptotically Euclidean or asymptotically hyperbolic CMC solutions are always nondegenerate (cf. §7 of [18]). In [19], using a definition of nondegeneracy similar to that stated above, we show how to glue non-CMC initial data sets, provided the data is CMC (same constant) near the gluing points. The argument from [19] readily applies to similar sets of non-CMC data on manifolds with boundary which are nondegenerate in the sense of Definition 1, yielding the following; ˜ γ˜ , K; ˜ pa ) be a smooth, marked solution of the Einstein vacuum Theorem 2.2. Let (M, ˜ an n-manifold with constraint equations with cosmological constant = 0 on M, boundary. We assume that the solution is nondegenerate and that the mean curvature, τ = trγ K is constant in the union of small balls (of any radius) about the points pa , a = 1, 2. Then there is a geometrically natural choice of a parameter T and, for T
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P. T. Chru´sciel, J. Isenberg, D. Pollack
sufficiently large, a one-parameter family of solutions (MT , T , KT ) of the Einstein constraint equations with the following properties. The n-manifold MT is constructed from M˜ by adding a neck connecting the two points p1 and p2 . For large values of T , ˜ the Cauchy data ( T , KT ) is a small perturbation of the initial Cauchy data (γ˜ , K) away from small balls about the pointspa . In fact, for any > 0 and k ∈ N we have ˜ as T → ∞ in C k M \ (B(p1 , ) ∪ B(p2 , )) . ( T , KT ) → (γ˜ , K)
3. The (Global) Gluing Construction for the Einstein-Matter Constraints for Manifolds with boundaries We need to show that the gluing construction which we have just described for the Einstein vacuum constraints on a manifold with boundary can be extended to the case of the Einstein-matter constraints (1.4), including a cosmological constant . In [17], Isenberg, Maxwell and Pollack describe in detail how to carry out gluing constructions analogous to that of [18] for solutions of the constraints for Einstein’s theory coupled to a wide variety of source fields (Maxwell, Yang-Mills, fluids, etc.) on complete manifolds. Here, we briefly describe how this works, and we adapt these results to the case of a manifold with boundary. For present purposes, we ignore any extra constraints which might have to be satisfied by the matter fields, and we describe those fields exclusively in terms of their stress-energy contributions3 ρ and J i . These are required to satisfy the dominant energy condition (1.3). We also allow for the inclusion of a non-zero cosmological constant . ˜ γ˜ , K, ˜ ρ, ˜ which satisfies the constraint So we start with a set of initial data (M, ˜ J˜, ) ˜ an n-dimensional manifold with smooth non-empty boundequations (1.4)–(1.5) on M, ary. We presume that this set of data has constant mean curvature τ˜ , from which it follows ˜ the trace-free field that if we do a trace decomposition K˜ = ν˜ + 13 τ˜ γ˜ , with τ˜ = trγ˜ K, ν˜ satisfies the condition ∇˜ j ν˜ j i = 8π J˜i . Along with the initial data, we specify the pair of points p1 , p2 ∈ M at which we carry out the gluing. We recall that the first step of the gluing construction for the vacuum constraints in [18] involves a conformal blowup of the gravitational fields at each of the points p1 and p2 , followed by gluing these fields together using cutoff functions along the join of the two asymptotic cylinders created by this blowup. For the non-vacuum case, we need to conformally transform and glue the matter quantities ρ and J as well. The conformal transformations which keep the conformal constraints semi-decoupled for CMC data, which preserve the dominant energy condition, and which lead to the simplest form 2n+2 2n+4 for the Lichnerowicz equation, are ρ˜ → φ n−2 ρ˜ and J˜i → φ n−2 J˜i (coupled with 4 ˜ → ). ˜ As for the gluings, γ˜ij → φ − n−2 γ˜ij , ν˜ ij → φ 2 ν˜ ij , and τ˜ → τ˜ , together with 2n+2 2n+4 we also apply a simple cutoff function procedure to φ n−2 ρ˜ and φ n−2 J˜i , thereby producing the smooth fields ρ˜T and J˜T on MT , along with γ˜T , ν˜ T , and the constants τ = τ˜ ˜ and = . 3 This can be interpreted as a perfect fluid model. However, we are not making any hypotheses upon the dynamics of the matter fields.
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The next step is finding the traceless tensor σ˜ T which satisfies the momentum constraint (γT ) j i σ˜ T
∇j
= 8π J˜Ti .
(3.1)
Here ∇ (γT ) is the Levi-Civita covariant derivative of the metric γT . We obtain σ˜ T by solving the boundary value problem LX = V in MT , (3.2) X = 0 on ∂MT , with L as in Sect. 2, and with V = JT − divγT ν˜ T , and then setting σ˜ T = ν˜ T + DX (recall that D has been defined in the paragraph preceding (2.1)). Noting that V is supported near the points p1 and p2 , we readily verify that the arguments for solvability of the boundary value problem (2.1) in Sect. 2 apply here as well. We also obtain the required pointwise estimates for σ˜ T − ν˜ T . The remaining step in the gluing construction of [18] involves solving the Lichnerowicz equation and then obtaining the requisite estimates for the solution ψT (i.e., showing that away from the neck, ψT → 1 in a suitable sense). For the Einsteinmatter constraints (1.4)–(1.5), with the decompositions described here, the Lichnerowicz operator takes the form (compare (2.3)) −3n+2 n−2 n−2 RT ψ + |σ˜ T |2 ψ n−2 4(n − 1) 4(n − 1) 1 n+2 n 4π(n − 2) 1 + ρT ψ − n−2 − (n − 2) τ2 − ψ n−2 . (3.3) n−1 4n 2(n − 1)
NT (ψ) = T ψ −
The matter-related term in (3.3), 2πρT ψ −3 , causes very few changes in the analysis. We note, for example, that in the expression for the linearised Lichnerowicz operator, LT = γT −
n−2 4(−n+1) −3n + 2 R(γT ) + |σ˜ T |2 ψT n−2 4(n − 1) 4(n − 1) 1 4 4πn 1 − 2n−2 − ρT ψT n−2 + (n + 2) τ2 − ψTn−2 , n−1 4n 2(n − 1)
the ρ term has very much the same effect as does the σ term, so its presence does not alter the proof of the existence of a solution or the subsequent error analysis. The constant , on the other hand, can cause trouble. However the argument presented in Sect. 2 shows that LT has a uniformly bounded inverse for T sufficiently large provided that (trg K)2 ≥
2n . (n − 1)
(3.4)
If this condition holds, then the rest of the analysis goes through. We thus have, finally Theorem 3.1. Let (M, γ , K, ρ, J, ; p1 , p2 ) be a smooth, marked, constant mean curvature solution of the Einstein matter constraint equations on M, an n-manifold with
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boundary. We assume that (3.4) holds, and that the dominant energy condition ρ ≥ |J | holds. Then there is a geometrically natural choice of a parameter T and, for T sufficiently large, a one-parameter family of solutions (MT , T , KT , ρT , JT , ) of the Einstein constraint equations with the following properties. The n-manifold MT is constructed from M by adding a neck connecting the two points p1 and p2 . For large values of T , the Cauchy data ( T , KT , ρT , JT , ) is a small perturbation of the initial Cauchy data (γ , K, ρ, J, ) away from small balls about the points p1 , p2 . In fact, for any > 0 and k ∈ N we have ( T , KT , ρT , JT , ) → (γ , K, ρ, J, ) as T → ∞ in C k M \ (B(p1 , ) ∪ B(p2 , )) . We note, without further discussion, that one can also readily produce a theorem analogous to Theorem 2.2, but with matter included in the constraint equations. 4. Proof of Theorems 1.1 and 1.4 In the vacuum case let (M , g) be the maximal globally hyperbolic vacuum development ˜ γ˜ , K); ˜ in the non-vacuum case let (M , g) be the development of of the initial data (M, the data, the existence of which has been assumed. In the vacuum case M˜ is achronal in M by construction; in the non-vacuum case this can be achieved, without loss of generality, by passing to a subset of M . There exists r0 > 0 such that for all 0 < r ≤ r0 , the open geodesic balls B(pa , r) ˜ γ˜ ) have smooth boundaries and relatively compact domains of dependence in in (M, (M , g). In the setting of Theorem 1.4 we set a = B(pa , r0 ). By reducing r0 if necessary we can assume that ρ > |J | on the domains of dependence D( a ). Without loss of generality we can further assume that r0 ≤ /2, where is the radius chosen in the statement of the theorems. By a result in[7], we can make an -small deformation of the initial data, supported in 1 ∪ 2 , such that the deformed initial data set satisfies the dominant energy condition, remains vacuum if it was to begin with, still satisfies K (B(pa , r0 )) = {0}, and now moreover there exists an R such that for every r− and r+ satisfying 0 < r− < r+ < R < r0 , we have K ( (pa , r− , r+ )) = {0},
(4.1)
where (pa , r− , r+ ) := B(pa , r+ )\B(pa , r− ). (In fact, the deformation can be arranged so that K (U ) = {0} for any open set U ⊂ B(pa , r0 ).) In vacuum, replacing a with B(pa , r0 ) if necessary, we may work in B(pa , r0 ) with r0 being taken as small as desired. We assume in what follows that this is the case. For any set with a distance function d, we define (s) := {p ∈ : d(p, ∂ ) < s} ; the sets considered here will always be equipped with a Riemannian metric, and then d will be taken to be the distance function associated with this metric. In particular we thus have a (s) = (pa , r0 − s, r0 ). Let us denote by (γa , Ka ) the initial data induced on a . We next wish to reduce the problem to that in which ( a , γa , Ka ) have constant (sufficiently large) mean curvature. We choose a constant τ so that τ2 −
2n ≥0. (n − 1)
(4.2)
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As the domains of dependence D( a ) are compact, we can use the work of Bartnik a ⊂ M , [2, Theorem 4.1] to conclude that there exist smooth spacelike hypersurfaces with boundaries ∂ a , on which the induced data (γa , Ka ) satisfy trγa Ka = τ .
(4.3)
In the Einstein matter case, with ρ > |J |, we appeal to the results in [7] to obtain a , preserving (4.3), such that there are no a small perturbation of the data induced on a . By continuity the dominant energy condition KIDs on any open subset of the regions ρ > |J | will still hold provided the perturbation is small enough. a have no local KIDs on every In the vacuum case, we claim that the domains collar neighborhood of their boundary. Indeed, suppose that this is not the case. Then 1 (s) ⊂ 1 , with a non-trivial set of KIDs there. there exists a collar neighborhood, say Therefore there exists a non-trivial Killing vector field X on the domain of dependence 1 (s)). But the intersection D( 1 (s)) ∩ 1 D( contains some collar neighborhood 1 (s1 ), and therefore X induces a KID there, contradicting (4.1). For all s0 > 0 the argument just given also guarantees the existence of an s1 satisfying 0 < s1 < s0 such that a (s0 ) \ a (s1 )) = {0} . K (
(4.4)
The process described so far reduces the problem to one with CMC initial data satisfying (4.2)–(4.4), on a compact manifold with boundary. (As pointed out in the introduction, the hypothesis of existence of the associated space-time, made in Theorem 1.4, a \ ∂ a . Applying is not needed for such data.) Choose now a pair of points pˆ a ∈ a , γa , Ka ) for any sufficiently Theorem 2.1 in vacuum or Theorem 3.1 with matter to ( γ (), K()), where M is the manifold, small , we obtain a glued initial data set (M, 2 across a with boundary ∂ M = ∂ 1 ∪ ∂ 2 , which is the connected sum of 1 and a (s0 ). On small neck around the points pˆ a . Let s0 > 0 be any number such that pˆ a ∈ a (s0 ) the deformed data (γ (), K()) arising from Theorem 2.1 approach (γa , Ka ) in any C k norm as goes to zero. As a consequence of (4.4), the construction presented in Sect. 8.6 of [12] can be carried through at fixed ρ and J i and it gives, for all small which coincides with (γa , Ka ) enough, a smooth deformation of (γ (), K()) on M, a (s0 ). The deformation preon a (s1 ), and coincides with (γ (), K()) away from serves the strict dominant energy condition (reducing if necessary), or is vacuum if the original data were. Consider, finally, the manifold M which is obtained by gluing together M˜ \( 1 ∪ 2 ) across ∂ M. M carries an obvious initial data set (γ , K), which is smooth except and M, at which both γ and some components of K are at perhaps at the gluing boundary ∂ M, bounded away from the neck region, least continuous. But in a neighborhood of ∂ M, the data (γ , K) coincide with those arising from a continuous, piecewise smooth hyper with a on the surface in M , which consists of a gluing of M˜ on one side of ∂ M, other. If we smooth out that hypersurface in M around ∂ a , then the new data near ∂ a arising from the smoothed-out hypersurface, provides a smoothing of the initial data constructed so far.
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5. Applications 5.1. Vacuum space-times without CMC surfaces. In [3] Bartnik has constructed an inextendible spatially compact space-time, satisfying the dominant energy condition, which has no closed CMC hypersurfaces (see also [11, 19, 22]). Here, using a construction analogous to that proposed by Eardley and Witt [15], we prove a similar result (Corollary 1.3) for vacuum spacetimes. The key step is proving the existence of vacuum initial data on a connected copy of T 3 with itself, with the property that the metric is symmetric under a reflection across the middle of the connecting neck, while K changes sign under this reflection. The non-existence of closed CMC surfaces in the maximal globally hyperbolic development of those initial data follows then from the arguments presented in [3]. Let γˆ be any metric on M = T 3 which has no conformal Killing vectors (such metrics exist, e.g. by [7]), let µˆ ≡ 0 be any transverse traceless tensor on M (such tensors exist, e.g. by [8]), and let Kˆ = µˆ + τ γˆ , for some constant τ = 0. It follows, e.g. from [16], ˜ with γ˜ that the conformal method applies, leading to a vacuum initial data set (γ˜ , K), ˜ being a conformal deformation of γˆ . Now, it is easily checked that for CMC data (γ˜ , K) on a closed manifold, a KID (N, Y ) must have N = 0 and must have Y a Killing vector field of γ˜ . Consequently, Y is a conformal Killing vector for γˆ , so that Y = 0 by our ˜ does not have any nontrivial global KIDs; i.e., hypothesis on γˆ . It follows that (γ˜ , K) K (M) = {0}. Let (M , g) be the maximal globally hyperbolic development of the data. As in Sect. 4, we can deform the initial data hypersurface in M to create a small neighborhood of a point p in which the trace of the new induced K˜ vanishes, while maintaining the condi˜ to denote the new data. tion K (M) = {0}. We use the same symbols (γ˜ , K) ˜ on the first copy, Now let M˜ consist of two copies of M, with initial data (γ˜ , K) ˜ on the second copy, say M2 . We let a = Ma , and say M1 , and with data (γ˜ , −K) we let pa denote the points in Ma corresponding to p. Noting that the mean curvature vanishes in symmetric neighborhoods of p1 and p2 , we now apply the construction for Theorem 1.1 presented in Sect. 4. To produce the desired initial data set on T 3 #T 3 , it is crucial to verify that all the steps are done with the correct symmetry around the middle of the connecting neck. In particular, we must check that the glued solution obtained from Theorem 2.1, when applied to such initial data, leads to a solution of the constraints which has the desired symmetry: this is achieved by using approximate solutions with the correct symmetry in the construction used to prove Theorem 2.1. The end result follows from the uniqueness (within the given conformal class) of the solutions obtained there. We thus have verified Corollary 1.3.
5.2. Bray’s quasi-local inner mass. In [9] Bray defines a notion of “inner mass” for a surface which is outer-minimising with respect to area in an asymptotically Euclidean initial data set (M, γ , K) satisfying the dominant energy condition (1.3) (see also [10]). Given a surface ⊂ M which is outer-minimising with respect to a fixed asymptotically flat end of (M, γ ), define the region I ‘inside’ to be the union of the components of M \ containing all the ends of M except √ for the chosen one. The inner mass minner () is then defined to be the supremum of A/16π taken over all fill-ins of (or replacements of I ⊂ M with initial data sets (of arbitrary topology) which satisfy (1.3) and extend smoothly to M \ I , with the data on M \ I unchanged) where A is the minimum area of surfaces in the fill-in needed to enclose all the ends other than the chosen
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‘exterior’ end. Note the similarity of this definition to Bartnik’s notion of quasi-local mass [4, 5]. It is by no means clear that extremal data which realise minner () exist. However Bartnik has observed that the construction described in this paper results in the following: Theorem 5.1. Suppose that (M, γ , K) is an asymptotically flat initial data set which realises the inner mass minner () for an outer minimising surface ⊂ M. Thus there is a surface (not necessarily connected) S ⊂ I , the interior region of (M, γ , K) relative to √ , such that A = Area(S) satisfies minner () = A/16π . If there is an open set ⊂ I satisfying ∂ = S ∪ , then there is at least one non-trivial KID on i.e. K ( ) = {0}. In particular, in the time-symmetric case, K ≡ 0, the resulting vacuum space-time is static in the domain of dependence of . The proof of Theorem 5.1 is an immediate consequence of the fact that were there to be no KIDS on , K ( ) = {0}, we could apply Theorem 1.1 and locally glue in an additional black hole whose apparent horizon would contribute an additional area to A. This would contradict the assumption that the original data was extremal for the inner mass. Acknowledgements. We acknowledge support from the Centre de Recherche Mathématiques, Université de Montréal, and the American Institute of Mathematics, Palo Alto, where the final stages of work on this paper were carried out. PTC and JI are also grateful to the Mathematics Department of the University of Washington and its members for their friendly hospitality.
References 1. Andersson, L., Chru´sciel, P.T.: On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”. Dissert. Math. 355, 1–100 (1996) 2. Bartnik, R.: Regularity of variational maximal surfaces. Acta Math. 161, 145–181 (1988) 3. Bartnik, R.: Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys. 117, 615–624 (1988) 4. Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62, 2346–2348 (1989) 5. Bartnik, R.: Energy in general relativity, Tsing Hua Lectures on Geometry and Analysis (S.-T. Yau, ed.), Cambridge, MA: International Press, 1997 6. Beig, R., Chru´sciel, P.T.: Killing Initial Data. Class. Quantum. Grav. 14, A83–A92 (1996). A special issue in honour of Andrzej Trautman on the occasion of his 64th Birthday, Tafel, J. (ed.). 7. Beig, R., Chru´sciel, P.T., Schoen, R.: KIDs are non-generic. To appear in Ann. Henri Poincaré; http://arxiv.org/list/gr-qc/0403042, 2004 8. Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Diff. Geom. 3, 379–392 (1969) 9. Bray, H.L.: Proof of the Riemannian Penrose conjecture using the positive mass theorem. J. Diff. Geom. 59, 177–267 (2001). 10. Bray, H.L., Chru´sciel, P.T.: The Penrose inequality. In: 50 years of the Cauchy problem in general relativity, Chru´sciel, P.T., Friedrich, H. eds., Basel: Birkhaeuser, 2004 11. Brill, D.: On spacetimes without maximal surfaces. In: Proceedings of the third Marcel Grossmann meeting (Amsterdam), Hu Ning, ed., Amsterdam: North Holland, 1983, pp. 79–87 12. Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. de France. 94, 1–103 (2003) 13. Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000) 14. Corvino, J., Schoen, R.: On the asymptotics for the vacuum Einstein constraint equations. To appear J. Diff. Geom.; http://arxiv.org/list/gr-qc/0301071, 2003 15. Eardley, D., Witt, D.: Unpublished, 1992 16. Isenberg, J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Grav. 12, 2249–2274 (1995)
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17. Isenberg, J., Maxwell, D., Pollack, D.: A gluing construction for non-vacuum solutions of the Einstein constraint equations. http://arxiv.org/list/gr-qc/0501083, 2005 18. Isenberg, J., Mazzeo, R., Pollack, D.: Gluing and wormholes for the Einstein constraint equations. Commun. Math. Phys. 231, 529–568 (2002) 19. Isenberg, J., Mazzeo, R., Pollack, D.: On the topology of vacuum spacetimes. Annales Henri Poincaré 4, 369–383 (2003) 20. Joyce, D.: Constant scalar curvature metrics on connected sums. Int. J. Math. Sci. no. 7, 405–450 (2003) 21. Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations I. J. Math. Phys. 16, 493–498 (1975) 22. Witt, D. M.: Vacuum space-times that admit no maximal slice. Phys. Rev. Lett. 57, 1386–1389 (1986) Communicated by G.W. Gibbons
Commun. Math. Phys. 257, 43–50 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1346-1
Communications in
Mathematical Physics
Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes Antonio N. Bernal, Miguel S´anchez Dpto. de Geometr´ıa y Topolog´ıa, Facultad de Ciencias, Fuentenueva s/n, 18071 Granada, Spain Received: 28 January 2004 / Accepted: 22 October 2004 Published online: 15 April 2005 – © Springer-Verlag 2005
To Professor P.E. Ehrlich, wishing him a continued recovery and good health Abstract: The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function T whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting M = R × S, g = −β(T , x)dT 2 + g¯ T , (b) if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function T with timelike gradient ∇T on all M. 1. Introduction The present article deals with some folk questions on differentiability of time functions and Cauchy hypersurfaces, as a natural continuation of our previous paper [2]. The following questions have been widely controversial since the very beginning of Causality Theory (see [2, Sect. 1] for a discussion and references1 ): (i) must any globally hyperbolic spacetime contain a smooth spacelike Cauchy hypersurface? [6, p. 1155], (ii) can classical Geroch’s topological splitting of globally hyperbolic spacetimes [3] be strengthened in a smooth orthogonal splitting?, and (iii) does any stably causal spacetime admit a smooth function with timelike gradient on all M? [1, p. 64]. The first question was answered affirmatively in [2], and our aim is to answer the other two. Concretely, for question (ii) we prove: Theorem 1.1. Let (M, g) be a globally hyperbolic spacetime. Then, it is isommetric to the smooth product manifold R × S,
·, · = −β dT 2 + g, ¯
The second-named author has been partially supported by a MCyT-FEDER Grant, MTM200404934-C04-01. 1 See also the authors’ contribution to Proc. II Int. Meeting on Lorentzian Geometry, Murcia, Spain, 2003, Publ. RSME vol. 8 (2004) 3–14, gr-qc/0404084.
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where S is a smooth spacelike Cauchy hypersurface, T : R × S → R is the natural projection, β : R × S → (0, ∞) a smooth function, and g¯ a 2-covariant symmetric tensor field on R × S, satisfying: 1. ∇T is timelike and past-pointing on all M (in particular, T is a time function). 2. Each hypersurface ST at constant T is a Cauchy hypersurface, and the restriction g¯ T of g¯ to such a ST is a Riemannian metric (i.e. ST is spacelike). 3. The radical of g¯ at each w ∈ R × S is Span∇T (=Span ∂T ) at w. For question (iii), recall first that a stably causal spacetime M is a causal spacetime which remains causal when its metric is varied in some neighborhood for the C 0 topology of metrics [1, p. 242]. It is well-known from the causal ladder of spacetimes that any globally hyperbolic spacetime is stably causal, but the converse does not hold [1, p. 73]; even more, stably causal spacetimes may fail to be causally continuous ([1, p. 71], see also [7] for detailed proofs and discussions on causally continuous and stably causal spacetimes). Hawking [4] proved that any stably causal spacetime admits a time function, i.e., a continuous function t : M → R which is strictly increasing on any future–directed causal curve. In fact, causally continuous spacetimes are characterized as spacetimes such that the past and future volume functions are time functions (for one, and then for any, admissible Borel measure). In the case of stably causal spacetimes, a time function is obtained as an appropiate “average” of volume functions for causal metrics obtained by widening the cones of the original one. Nevertheless, even though the continuity of such an average function is proved, its smoothability remained as a “folk problem” (see also [5, Prop. 6.4.9] , [1, Sect. 3.2], [7, Sect. 4]). Conversely, Hawking also proved that a spacetime is stably causal if it admits a smooth function with everywhere timelike gradient. We will call such a function a temporal function, i.e., a smooth function T on a spacetime M with (past-directed) timelike gradient ∇T on all M. Notice that, obviously, any temporal function is a time function, but even a smooth time function may be non-temporal (it may have a lightlike gradient in some points). Our technique also proves: Theorem 1.2. Any spacetime M which admits a time function also admits a temporal function. This result, combined with Hawking’s ones, ensures, on one hand, that any stably causal spacetime admits a temporal function and, on the other, that any spacetime which admits a time function is stably causal. Notice that, in fact, Theorem 1.1 ensures the existence of a Cauchy temporal function T in any globally hyperbolic spacetime, i.e., a temporal function with Cauchy hypersurfaces as levels. So, the proof of Theorem 1.2 can be carried out easily by simplifying the reasonings for Theorem 1.1 (the property of being Cauchy is not taken into account now). For the proof of this last theorem, the reader is assumed to be familiarized with the technique in [2]. Our approach is very different from previous ones on this topic. Essentially, the idea goes as follows. Let t be a continuous Cauchy time function as in Geroch’s theorem. As shown in [2], if t− < t then there exists a smooth Cauchy hypersurface S contained in t −1 (t− , t); this hypersurface is obtained as the regular value of a certain function with either timelike or zero gradient on t −1 (t− , t]. As t− approaches t, S can be seen as a smoothing of St ; nevertheless S always lies in I − (St ). In Section 2 we show how the required splitting of the spacetime would be obtained if we could ensure the existence
Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes
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of a temporal step function τ around each St . Essentially, such a τ is a function with timelike gradient on a neighborhood of St (and zero gradient outside) with level Cauchy hypersurfaces which cover a rectangular neighbourhood of St (Definition 2.3). Then, Sect. 3 is devoted to prove the existence of such a temporal step function around any St . To this aim, we will show first how St (but perhaps no other close Cauchy hypersurface obtained varying t) can be covered by Cauchy level hypersurfaces of a certain function τˆ , Proposition 3.6. Then, the temporal stepfunction τ will be obtained as the sum of a series of previously constructed functions j τˆ [j ] , Theorem 3.11, and special care will be necessary to ensure its smoothness. Finally, in Remark 3.12 we sketch Theorem 1.2. Essentially, only some variations on previous arguments are needed, and we also sketch a proof by taking into account that each level hypersurface S of the continuous time function t is not only achronal, but also a Cauchy hypersurface in its Cauchy development D(S). 2. Temporal Step Functions In what follows, M ≡ (M, g) will denote a n-dimensional globally hyperbolic spacetime, and t a continuous Cauchy time function given by Geroch’s theorem, as in [2, Prop. 4]. Then, each St = t −1 (t) is the corresponding topological Cauchy hypersurface, and the associated topological splitting is M = R × S, where S is any of the St ’s (see [2, Prop. 5]). N = {1, 2, . . . } will denote the natural numbers (Z, R, resp., integers, real numbers). In principle, “smooth” means C r -differentiable, where r ∈ N ∪ {∞} is the maximum degree of differentiability of the spacetime. Nevertheless, we will assume r = ∞, and the steps would remain equal if r < ∞, except for some obvious simplifications in the proof of Theorem 3.11. W will denote the topological closure of the subset W ⊂ M. Definition 2.3. Given the Cauchy hypersurface S ≡ St , fix t− , t+ , ta , tb ∈ R, t− < ta < t < tb < t+ , and put S− = St− , S+ = St+ , It = (ta , tb ). We will say that τ :M→R is a temporal step function around t, compatible with the outer extremes t− , t+ and the inner extremes ta , tb , if it satisfies: 1. ∇τ is timelike and past-pointing where it does not vanish, that is, in the interior of its support V := Int(Supp(∇τ )). 2. −1 ≤ τ ≤ 1. 3. τ (J + (S+ )) ≡ 1, τ (J − (S− )) ≡ −1. In particular, the support of ∇τ satisfies: Supp(∇τ ) ⊂ J + (S− ) ∩ J − (S+ )(= t −1 [t− , t+ ]). 4. St ⊂ V , for all t ∈ It ; that is, the rectangular neighborhood of S, t −1 (It ) ≡ It × S, is included in V (or J + (Sta ) ∩ J − (Stb ) ⊂ Supp(∇τ )). Recall that, from the first property, the inverse image of any regular value of τ is a smooth closed2 spacelike hypersurface. Even more, from the third property such hypersurfaces are Cauchy hypersurfaces (use [2, Cor. 11]) and, from the fourth, they cover not only St but also close Cauchy hypersurfaces St . 2 Here, closed hypersurface means closed as a subset of M (but not necessarily compact); hypersurfaces are always assumed embedded without boundary.
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Proposition 2.4. Assume that the globally hyperbolic spacetime M admits a temporal step function around t, for any t ∈ R, compatible with outer extremes t+ = t + 2, t− = t − 2. Then, there exists a smooth function T : M → R which satisfies: (A) ∇T is timelike and past-pointing on all M. (B) For each inextendible timelike curve γ : R → M, parameterized with t, one has: limt→±∞ (T (γ (t)) = ±∞. Thus, each hypersurface at constant T , ST = T −1 (T ), is a smooth spacelike Cauchy hypersurface, and all the conclusions of Theorem 1.1 hold. Proof. Consider, for each t ∈ R, the function τ ≡ τt , the open subset Vt and the interval It in Definition 2.3, and, thus, the open covering of M, V = {Vt : t ∈ R}, with associated open covering I = {It , t ∈ R} of R. As the length of each It is < t+ − t− = 4, a locally finite subrecovering I of I exists (we can also assume that no interval in this subrecovering is included in another interval of it) and, as a consequence, a locally finite subrecovering V of V: V = {Vtk : k ∈ Z}. Without loss of generality, we can assume tk < tk+1 and then, necessarily, lim tk = ±∞.
k→±∞
(1)
The notation will be simplified Vk ≡ Vtk , τk ≡ τtk . Define now: T = τ0 +
∞
(τ−k + τk ).
(2)
k=1
Notice that τ−k +τk ≡ 0 on J + (St−k +2 )∩J − (Stk −2 ), for all k (this applies when k > k0 , where k0 is the first k > 0 such that t−k + 2 < tk − 2). This (plus the limit (1)) ensures that T is well defined and smooth. Property (A) is then straightforward from the definition of T , the convexity of the (past) time cones and the fact that V covers all M. For (B), consider the limit to +∞ (to −∞ is analogous). It is enough to check that, for any k ∈ N there exists t k ∈ R such that T (γ (t k )) > k (and, thus, from (A), this inequality holds for all t > t k ). But taking t k = tk + 2 (≥ Sup(t (Vk ))), one has T (γ (t k )) > 2k obviously from (2). To check that the (necessarily smooth and spacelike) hypersurface ST is Cauchy, notice that no timelike curve (in fact, no causal one) can cross ST more than once because of property (A). Thus, any inextendible timelike curve γ can be T -reparameterized in some interval (T− , T+ ) and, because of (B), necessarily T± = ±∞. Therefore, γ must cross each ST . Now, the assertions in Theorem 1.1 are straightforward consequences of previous properties. Briefly, let S = T −1 (0) and define the map : M → R × S,
p → (T (p), (p)),
where (p) is the unique point of S crossed by the inextendible curve of ∇T through p. The vector field ∂/∂T obtained at each point p as the derivative of the curve
Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes
47
s → −1 (T (p) + s, (p)) is clearly colinear to ∇T at each point. Even more, as g(∂/∂T , ∇T ) ≡ 1, then ∂/∂T = −∇T /|∇T |2 . Thus, the metric ∗ g induced on R × S satisfies all the required properties with β((p)) = |∇T |−2 (p), for all p ∈ M. Remark 2.5. (i) The restriction on the outer extremes can be obviously weakened by assuming that t+ − t− is bounded. Thus, Proposition 2.4 reduces Theorem 1.1 to prove the existence of a temporal step function around any St with bounded outer extremes. Theorem 3.11 will prove this result. (ii) In fact, Theorem 3.11 proves more: the outer and inner extremes can be chosen arbitrarily. Thus, one can assume always for temporal step functions tb = t + 1, ta = t − 1, t+ = t + 2, t− = t − 2. In this case, the proof of Proposition 2.4 can be simplified because one can take directly the subrecovering V with tk = k for all k ∈ Z. 3. Construction of a Temporal Step Function Proposition 3.6. For each S ≡ St there exists a function τˆ ≡ τˆt which satisfies the three first properties in Definition 2.3 and, additionally: ˆ S ⊂ V. 4. For its proof, we will need first the following two lemmas, which are straightforward from [2]. Thus, we only sketch the steps for their proofs. Lemma 3.7. Let S ≡ St be a Cauchy hypersurface. Then there exists an open subset U with J − (S) ⊂ U ⊂ I − (St+1 ), and a function h+ : M → R, h+ ≥ 0, with support included in I + (St−1 ) which satisfies: (i) If p ∈ U with h+ (p) > 0 then ∇h+ (p) is timelike and past-pointing. (ii) h+ > 1/2 (and, thus, its gradient is timelike past pointing) on J + (S) ∩ U . Proof. Recall that the function h constructed in [2, Prop. 14] (putting S = S2 , St−1 = S1 ) yields directly h+ . In fact, function h in that reference also satisfies (ii) in an open neighborhood U of S (this is obvious because that function h is constructed from the sum of certain functions hp in [2, Lemma 5] which satisfy (ii) on appropiate open subsets which cover S), which can be chosen included in I − (St+1 ), and, thus, take U = U ∪ I − (S). Lemma 3.8. Let S be a Cauchy hypersuperface and U (⊂ I − (St+1 )) an open neighborhood of J − (S). Then, there exists a function h− : M → R, h− ≤ 0, with support included in U satisfying: (i) If ∇h− (p) = 0 at p (∈ U ) then ∇h− (p) is timelike past-pointing. (ii) h− ≡ −1 on J − (S). Proof. The proof would follow as the construction of h in [2, Prop. 14] with the following modifications: (a) reverse the time-orientation, and consider [2, Prop. 14] with S = S2 , St+1 = S1 , (b) take all the convex open subsets Cp included in U , (c) construct h by exactly the same method, but changing the sign of all the Lorentz distances (i.e., time–separations are taken negative), and (d) once h ≤ 0 is constructed in this way
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(notice that h(S) < −1/2), define h− on J + (S) (with the original time-orientation in what follows) as: h− = ϕ ◦ h, where ϕ : R → R is any function which satisfies ϕ ([−1/2, 0]) > 0,
ϕ((−∞, −1/2]) ≡ −1,
(of course, h− is defined on J − (S) as equal to −1).
ϕ(0) = 0
Proof of Proposition 3.6. Fixed St , take U , h+ and h− as in the two previous lemmas. Notice that h+ − h− > 0 on all U . Then, define: τˆt = 2
h+ −1 h+ − h −
on U , and constantly equal to 1 on M\U . As ∇ τˆt = 2
h+ ∇h− − h− ∇h+ (h+ − h− )2
is either timelike or 0 everywhere, all the required properties are trivially satisfied.
We can even strengthen technically the conclusion of Proposition 3.6 for posterior referencing: Corollary 3.9. Let t− < ta < t < tb < t+ and a compact subset K ⊂ t −1 ([ta , tb ]). Then, there exists a function τˆ which satisfies the four properties of Proposition 3.6 and, additionally: K ⊂ V . Proof. For each St with t ∈ [ta , tb ], take the corresponding function τˆt from Proposition 3.6. K is then covered by the corresponding open subsets V t and, from compactness, a finite set of t’s, say, t1 , . . . , tm suffices. Then take τˆ = m−1 i τˆti . Theorem 1.1 will be the obvious consequence of Proposition 2.4 and Theorem 3.11 below. For the proof of this one, we will sum an appropiate series of functions as the ones in Corollary 3.9, and we will have to be careful with the smoothness of the sum. But, first, the following trivial lemma will ensure that the infinite sum will not be an obstacle for the timelike character of the gradient. be a sequence of timelike vectors in the same cone of a vector Lemma 3.10. Let {vi } space. If the sum v = ∞ i=1 vi is well defined then the vector v is timelike. Proof. As the causal cones are closed, ∞ i=2 vi is causal and, as the sum of a causal plus a timelike vector in the same cone is timelike, v = v1 + ∞ i=2 vi is timelike. Theorem 3.11. For each S ≡ St and t− < ta < t < tb < t+ there exists a temporal step function τ around S with outer extremes t− , t+ and inner extremes ta , tb , It = (ta , tb ).
Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes
49
Proof. Choose a sequence {Gj : j ∈ N} of open subsets such that: Gj is compact, Gj ⊂ Gj +1
M = ∪∞ j =1 Gj ,
(3)
and the associated sequence of inner compact subsets Kj = Gj ∩ J + (Sta ) ∩ J − (Stb ). For each Kj , consider the function τˆ [j ] given by Corollary 3.9 with K = Kj , and put Vj :=Int(Supp ∇ τˆ [j ] ), Kj ⊂ Vj . Notice that the series ∞ 1 [j ] τ˜ := τˆ , 2j
(4)
j =1
converges at each q ∈ M and, thus, defines a continuous function τ˜ : M → R. If τ˜ were smooth and its partial derivatives (in coordinate charts) commuted with the infinite , then τ˜ would be the required temporal step function, obviously (use Lemma 3.10). As these hypotheses have not been ensured, expression (4) will be modified as follows. Fix a locally finite atlas A = {Wi : i ∈ N} such that each chart W ≡ (W, x1 , . . . , xn ) ∈ A has a relatively compact domain and it is also the restriction of a bigger chart on M whose domain includes W . Then, each compact subset Gj is intersected by a finite number of neighborhoods Wi1 , . . . , Wikj . As D := (Wi1 ∪ · · · ∪ Wikj ) is compact, there exists Aj > 1 such that |τˆ [j ] | < Aj on D and, for each s < j : ∂ s τˆ [j ] ∀q ∈ D, ∀l1 , . . . , ls ∈ {1, . . . , n}. ∂x ∂x ...∂x (q) < Aj , l1 l2 ls Now, the series τ ∗ :=
∞ j =1
1 2j A
j
τˆ [j ] ,
(5)
is smooth on all M. In fact, to check differentiability C s at p ∈ M, choose j0 ∈ N and W ∈ A with p ∈ Gj0 ∩ W . Recall that, for any j > Max{j0 , s}, the summand 2j1A τˆ [j ] j and all its partial derivatives in the local coordinates of W until order s, are bounded in absolute value by 1/2j on Gj0 ∩ W . Thus, the series (5) and the partial derivatives converge uniformly on a neighborhood of p, and the derivatives commute with on M. Therefore, τ ∗ satisfies trivially all the properties of a temporal step function in Definition 2.3 except, at most, the normalizations to 1 and −1 in the second and third ones. Instead, τ ∗ satisfies τ ∗ (J − (S− )) ≡ c− < 0, τ ∗ (J + (S+ )) ≡ c+ > 0. The required function is then τ = ψ ◦ τ ∗ , where the smooth function ψ : R → R satisfies ψ > 0, ψ(c− ) = −1, ψ(c+ ) = 1. Remark 3.12. As said in the Introduction, the proof of Theorem 1.2 can be carried out directly by simplifying previous reasonings. Concretely, property (B) of Proposition 2.4 is not needed now, and property (A) can be achieved from temporal step functions as in Definition 2.3, where each St is a level hypersurface of the time function. Alternatively, let t be a time function, choose p ∈ M and let S = t −1 (t (p)) be the level hypersurface of t through p. Then, S is closed, achronal and separates M, i.e.,
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M\S is the disjoint union of the open subsets, M+ := t −1 (t (p), ∞)(⊇ I + (S)) and M− := t −1 (−∞, t (p))(⊇ I − (S)). Even more, S is a Cauchy hypersurface of its Cauchy development D(S). Now, recall: (i) Any temporal step function τ on D(S) can be extended to all M by putting τ (M+ \D + (S)) ≡ 1, τ (M− \D − (S)) ≡ −1. Thus, ∇τ is: (a) either timelike or zero everywhere, and (b) timelike on a neighborhood of p. (ii) Given any compact subset G ⊂ M, a similar function τˆ , which satisfies not only (a) but also (b) for all p ∈ G, can be obtained as a finite sum of functions constructed in (i) (in analogy to Corollary 3.9). (iii) Choosing a sequence of compact subsets Gj as in formula (3), taking the corresponding function τˆ [j ] obtained in (ii), and summing a series in a similar way than in (5), the required temporal function is obtained. References 1. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry. Monographs Textbooks Pure Appl. Math. 202, New York: Dekker Inc., 1996 2. Bernal, A. N., S´anchez, M.: On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem. Commun. Math. Phys. 243, 461–470 (2003) 3. Geroch, R.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970) 4. Hawking, S.W.: The existence of Cosmic Time Functions. Proc. Roy. Soc. London, Series A 308, 433–435 (1969) 5. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. London-NewYork: Cambridge University Press, 1973 6. Sachs, R.K., Wu, H.: General Relativity and Cosmology. Bull. Amer. Math. Soc. 83(6), 1101–1164 (1977) 7. S´anchez, M.: Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch’s splitting. A revision. In: Proceedings of the 13th School of Differential Geometry, Sao Paulo, Brazil, 2004 (to appear in Matematica Contemporanea). Available at http://arxiv.org/list/gr-qc/0411143, 2004 Communicated by G.W. Gibbons
Commun. Math. Phys. 257, 51–85 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1260-y
Communications in
Mathematical Physics
Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators Guillaume James, Yannick Sire Math´ematiques pour l’Industrie et la Physique, UMR CNRS 5640, and D´epartement GMM, Institut National des Sciences Appliqu´ees, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. E-mail:
[email protected];
[email protected]
Received: 6 April 2004 / Accepted: 10 July 2004 Published online: 11 January 2005 – © Springer-Verlag 2005
Abstract: We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schr¨odinger equation (M. Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchg¨assner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchg¨assner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchg¨assner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schr¨odinger approximation.
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1. Introduction We consider a one-dimensional lattice of nonlinear oscillators described by the following system (Klein-Gordon system): d 2 xn + V (xn ) = γ (xn+1 + xn−1 − 2xn ), n ∈ Z, dτ 2
(1)
where xn is the displacement of the nth particle from an equilibrium position, the coupling constant γ is strictly positive and the on-site potential V is analytic in a neighborhood of x = 0 (with V (0) = 0, V (0) > 0). This system describes a chain of particles linearly coupled to their first neighbors, in the local anharmonic potential V . In this paper, we consider solutions of (1) satisfying xn (τ ) = xn−p (τ − T ),
(2)
for a fixed T ∈ R and p ≥ 1. The case when p = 1 in (2) corresponds to travelling waves. Solutions satisfying (2) for p = 1 consist of pulsating travelling waves, which are exactly translated by p sites after a fixed propagation time T and are allowed to oscillate as they propagate on the lattice. In particular, solutions of (1) having the form xn (τ ) = x(n − c τ, τ ) (x being T -periodic in its second argument) satisfy (2) under the condition c = p/T . A different situation arises when c and 1/T are incommensurate, since the solution is not exactly translated on the lattice after time T but is modified by a spatial shift. Solutions of type (2) having the additional property of spatial localization (xn (τ ) → 0 as n → ±∞) are known as exact travelling breathers (with commensurate velocity and frequency) and have been studied numerically in different systems. Approximate travelling breather solutions propagating on the lattice at a non constant velocity c have also drawn a lot of attention. They have been numerically observed in various one-dimensional nonlinear lattices such as Fermi-Pasta-Ulam lattices [43, 8, 37, 13], Klein-Gordon chains [9, 6] and the discrete nonlinear Schr¨odinger (DNLS) equation [12]. The two last models exhibit similar features in some regimes where the DNLS equation can be derived from the Klein-Gordon system using appropriate scalings [35]. Other references are available in the review paper [15]. One way of generating approximate travelling breathers consists of “kicking” static breathers consisting of spatially localized and time periodic oscillations (see the basic papers [44, 30, 15, 25, 5] for more details on these solutions). Static breathers are put into motion by perturbation in the direction of a pinning mode [6]. The possible existence of an energy barrier that the breather has to overcome in order to become mobile has drawn a lot of attention, see e.g. [9, 6, 13, 26] and the review paper [39]. It is a more delicate task to examine the existence of exact travelling breathers using numerical computations. Indeed, these solutions might not exist without being superposed on a small nonvanishing oscillatory tail which violates the property of spatial localization. This phenomenon is likely to occur since the existence of a nonvanishing oscillatory tail has been previously observed in some parameter regimes for solitary waves (spatially localized travelling waves) in Klein-Gordon chains [6]. Numerical results indicate similar phenomena for the propagation of kinks [10, 38, 4]. Fine analysis of numerical convergence problems also suggests that different nonlinear lattices do not support exact solitary waves or travelling breathers in certain parameter regimes [42, 3]. Nevertheless, several formal analytical methods have been used to obtain travelling breather solutions. On the one hand, approximate travelling breathers can be formally obtained via effective Hamiltonians, which approximately describe the motion of the breather center on the lattice, at a nonconstant velocity [31, 26]. On the other hand,
Travelling Breathers in Klein-Gordon Chains
53
multi-scale expansions provide evolution equations for the envelopes of well-prepared initial conditions corresponding to modulated plane waves. This approach has been used by Remoissenet for Klein-Gordon lattices [36] and yields the nonlinear Schr¨odinger (NLS) equation as a modulation equation. For good parameter values, the NLS equation admits solitons corresponding (at least formally) to travelling breather solutions of the original system, which propagate at a constant velocity (the group velocity of the wave packet). At the order of the NLS approximation, the linear dispersion is exactly balanced by the effect of nonlinear terms. The same approach has been used by Tsurui for the Fermi-Pasta-Ulam lattice [45]. For the Klein-Gordon system (and generalizations with anharmonic coupling), the validity of the nonlinear Schr¨odinger equation on large but finite time intervals has been proved recently by Giannoulis and Mielke [19]. It is a challenging problem to determine if these approximate solutions could constitute the principal part of exact travelling breather solutions of the Klein-Gordon system. This would imply that linear dispersion is balanced by nonlinear terms at any order in the above mentioned multi-scale expansion. This problem has been solved by Iooss and Kirchg¨assner in the case of travelling waves [22], where the phase velocity of the plane wave equals the group velocity of the wave packet. Travelling wave solutions of (1) (with p = 1 in (2)) are determined by the scalar advance-delay differential equation d 2 x1 + V (x1 ) = γ (x1 (τ − T ) − 2x1 + x1 (τ + T )). (3) dτ 2 Iooss and Kirchg¨assner have studied small amplitude solutions of (3) in different parameter regimes and have obtained in particular “nanopterons” consisting of a solitary wave superposed on an exponentially small oscillatory tail. The leading order part of these solutions (excluding their tail) coincides with approximate solutions obtained via the NLS equation. However, the more general case when phase and group velocities are different has remained open until now. More generally, different situations have been observed for the existence of exact travelling breathers in various simpler models. On the one hand, exact travelling breathers can be explicitly computed in the integrable Ablowitz-Ladik lattice [1], and other examples of nonlinear lattices supporting exact travelling breathers can be obtained using an inverse method [14]. On the other hand, travelling breather solutions of the Ablowitz-Ladik lattice are not robust under various non-Hamiltonian reversible perturbations as shown in [7]. The aim of our study is to clarify the existence question of exact travelling breather solutions in the Klein-Gordon lattice (1), in a case when the breather period and the inverse of its velocity are commensurate (we develop the results announced in [40]). For fixed p ≥ 2, problem (1)–(2) reduces to the p-dimensional system of advance-delay differential equations x1 V (x1 ) x2 (τ ) − 2x1 (τ ) + xp (τ + T ) .. .. .. . . . d2 (4) xn + V (xn ) = γ xn+1 (τ ) − 2xn (τ ) + xn−1 (τ ) . 2 dτ . . . . . .. . . xp V (xp ) x1 (τ − T ) − 2xp (τ ) + xp−1 (τ ) For the sake of simplicity we restrict ourselves to the case p = 2 in (4). The general case p ≥ 2 is analyzed in a work in progress. The latter is technically more difficult but the approach used in our paper works as well.
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We analyze small amplitude solutions of (4) (with p = 2) using the method developed by Iooss and Kirchg¨assner [22] in the context of travelling waves (see [20] for an application of this method to Fermi-Pasta-Ulam lattices). The method is based on a reduction to a center manifold in the infinite dimensional case as described in references [27, 33, 46]. System (4) is rewritten as a reversible evolution problem in a suitable functional space, and considered for parameter values (T , γ ) near a critical curve where the imaginary part of the spectrum consists of a pair of double eigenvalues and two pairs of simple ones. Close to this curve, the pair of double eigenvalues splits in two pairs of eigenvalues with opposite nonzero real parts, which opens the possibility of finding homoclinic solutions to 0. Near these parameter values, the center manifold theorem reduces the problem locally to a reversible 8-dimensional system of differential equations. Thanks to an appropriate choice of variables, the reduction procedure is similar to the case analyzed by Iooss and Kirchg¨assner [22]. However, the simplest homoclinic bifurcation yields in our case a higher-dimensional reduced system, with a supplementary pair of simple imaginary eigenvalues. The reduced system is put in a normal form which is integrable up to higher order terms. In some regions of the parameter space, the truncated normal form admits reversible homoclinic orbits to 0, which bifurcate from the trivial state and correspond to approximate solutions of (4). These approximate solutions coincide with spatially localized modulated plane waves obtained via the NLS equation. However, by analogy with results of Lombardi [28] we conjecture that these solutions do not generically persist when higher order terms are taken into account in the normal form. To make a more precise statement fix V (x) = 21 x 2 + αx 3 + βx 4 . We expect that a reversible solution of the reduced equation homoclinic to 0 and close to a small amplitude homoclinic orbit of the truncated normal form might only exist if (T , γ , α, β) is chosen on a discrete collection of codimension-m submanifolds of R4 (m > 0). The codimension depends on the number of pairs of purely imaginary eigenvalues (i.e. the number of resonant phonons) in our parameter regime and symmetry assumptions. In our case (with two pairs of purely imaginary eigenvalues, in addition to hyperbolic ones), we expect m = 2 when homoclinic orbits to 0 correspond to travelling breather solutions of (1)–(2) (with p = 2), and m = 1 when homoclinic orbits to 0 correspond to solitary waves (homoclinic orbits to 0 possess an additional symmetry in that case). For general parameter values, instead of homoclinic orbits to 0 one can expect the existence of reversible homoclinic orbits to exponentially small 2−dimensional tori, originating from the two additional pairs of simple purely imaginary eigenvalues. These solutions should constitute the principal part of exact travelling breather solutions of (1) superposed on a small quasi-periodic oscillatory tail. However, in order to obtain exact solutions one has to prove the persistence of the corresponding homoclinic orbits as higher order terms are taken into account in the normal form. This step is non-trivial and would require to generalize results of Lombardi [28] available when one pair of simple imaginary eigenvalues is removed. The most intricate part of the problem is to obtain a sharp (exponentially small) estimate of the minimal tail size of solutions. Another promising approach for obtaining such estimates is developed in the recent work of Iooss and Lombardi [23] on polynomial normal forms with exponentially small remainder for analytic vector fields. However the application of their theory to our situation would require several nontrivial extensions (to the (iω0 )2 iω1 iω2 resonance and to systems with an additional infinite-dimensional hyperbolic part).
Travelling Breathers in Klein-Gordon Chains
55
In this paper we prove the persistence of some homoclinic solutions in the case when the on-site potential V is even. Indeed, due to the additional invariance xn → −xn one can find solutions of (1)–(2) (with p = 2) satisfying xn (τ ) = −xn−1 (τ − T2 ). These solutions correspond to solutions of the normal form system possessing a particular symmetry. For the normal form restricted to the associated (6-dimensional) invariant subspace, results of Lombardi [28] are applicable since the linear part does not possess an extra pair of simple purely imaginary eigenvalues (the bifurcation corresponds to a pair of double eigenvalues and a pair of simple ones). As a result the full normal form admits homoclinic orbits to small periodic ones for near-critical parameter values (T , γ ). These solutions correspond to exact travelling breather solutions of (1) superposed on a small periodic oscillatory tail, which can be made exponentially small with respect to the central oscillation size. The minimal tail size should be generically nonzero for a given value of (T , γ ), but might vanish on a discrete collection of curves in the (T , γ ) parameter plane. As a consequence, in a given system (1) (with fixed coupling constant γ and symmetric on-site potential V ), exact travelling breather solutions decaying to 0 at infinity (and satisfying (2) for p = 2) might exist in the small amplitude regime, for isolated values of the breather velocity 2/T . We insist on the fact that our study is local, and analytical results for large amplitude solutions would be of interest. Results of this type exist for solitary waves or kinks in several one-dimensional nonlinear lattices (see [18, 17, 32, 41, 16]) but the problem is still open for large amplitude travelling breather solutions. The paper is organized as follows. In Sect. 2 we formulate (1)-(2) as an evolution problem in an infinite-dimensional Banach space. Sections 3 and 4 are devoted to the linearized problem (spectral study, optimal regularity result) and the reduction to a center manifold. In Sect. 5 we study the reduced equation and describe its small amplitude homoclinic solutions when higher-order terms are neglected. These terms are taken into account in the even-potential case. Section 6 describes the corresponding leading-order travelling breather solutions of the Klein-Gordon system, and exact solutions (with small oscillatory tails) in the case of even potentials.
2. Formulation of the Problem In this section, we formulate the initial problem (1)–(2) in an appropriate way. The case p = 2 in (2) leads to the following system: d2 dτ 2
x1 V (x1 ) x2 (τ ) − 2x1 (τ ) + x2 (τ + T ) + =γ . V (x2 ) x1 (τ ) − 2x2 (τ ) + x1 (τ − T ) x2
(5)
Note that travelling wave solutions of (1) satisfying xn (τ ) = xn−1 (τ − T /2) are particular solutions of (2) with p = 2. Consequently, the solutions considered in our case include those found by Iooss and Kirchg¨assner [22]. We shall analyze small amplitude solutions of (5) using the center manifold reduction method introduced by Iooss and Kirchg¨assner [22] in the context of reversible advancedelay differential equations. For this purpose, one has to make a convenient choice of variables which allows us to recover some essential estimates in their reduction process (optimal regularity result). We rescale (5) using t = Tτ and consider the new variable (u1 (t), u2 (t)) = (x1 (τ ), x2 (τ + T2 )). This yields
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G. James, Y. Sire
τ n−1 − ) if n is odd, T 2 n−1 τ xn (τ ) = u2 ( − ) if n is even. T 2
xn (τ ) = u1 (
With this change of variables, we have
u2 (t − 21 ) − 2u1 (t) + u2 (t + 21 ) d 2 u1 2 V (u1 ) 2 +T . = γT V (u2 ) dt 2 u2 u1 (t + 21 ) − 2u2 (t) + u1 (t − 21 )
(6)
(7)
Note that solutions of (7) with u1 = u2 correspond to travelling wave solutions of (1) satisfying xn (τ ) = xn−1 (τ − T2 ). As in [22] we set U = (u1 , u2 , u˙ 1 , u˙ 2 , X1 (t, v), X2 (t, v))T , where v ∈ [−1/2, 1/2] and X1 (t, v) = u1 (t +v), X2 (t, v) = u2 (t +v). We define the following trace operators: δ1/2 Xi (t, v) = Xi (t, 1/2), δ−1/2 Xi (t, v) = Xi (t, −1/2).
(8) (9)
Furthermore, we assume V analytic in a neighborhood of 0, with the following Taylor expansion at x = 0: V (x) =
1 2 a 3 b 4 x − x − x + h.o.t. 2 3 4
(10)
We can write the system (7) as an evolution problem dU = LU + F (U ) dt
(11)
with L given by
0 0 α L= 1 0 0 0
0 0 0 α1 0 0
1 0 0 0 0 0
0 0 0 1 0 0 0 0 α2 (δ1/2 + δ−1/2 ) , 0 α2 (δ−1/2 + δ1/2 ) 0 0 0 ∂v 0 0 ∂v
(12)
α2 = T 2 γ and α1 = −T 2 (1 + 2γ ). The nonlinear operator F is given by F (U ) = T 2 (0, 0, f (u1 ), f (u2 ), 0, 0)T
(13)
f (u) = au2 + bu3 + h.o.t.
(14)
and
We now write (11) in appropriate function spaces. For this purpose we introduce the Banach spaces H = R4 × (C 0 [−1/2, 1/2])2 ,
(15)
D = U ∈ R4 × (C 1 [−1/2, 1/2])2 /X1 (0) = u1 , X2 (0) = u2 .
(16)
Travelling Breathers in Klein-Gordon Chains
57
The operator L maps D into H continuously, F : D → D is C k−1 with F (U ) = O(U 2D ). We observe that the symmetry R on H defined by R(u1 , u2 , ξ1 , ξ2 , X1 (v), X2 (v))T = (u1 , u2 , −ξ1 , −ξ2 , X1 (−v), X2 (−v))T satisfies (L + F ) ◦ R = −R(L + F ). Therefore, if U is a solution of (11) then R U (−t) is also a solution, i.e. the system (11) is reversible under R. This property is due to the invariance t → −t of (7). A solution U of (11) is said to be reversible under R if R U (−t) = U (t) for all t ∈ R. Reversible solutions under R correspond to solutions of (1)–(2) satisfying x−n (−τ − T ) = xn (τ ). In addition, note that the permutational symmetry S(u1 , u2 , ξ1 , ξ2 , X1 , X2 )T = (u2 , u1 , ξ2 , ξ1 , X2 , X1 )T
(17)
commutes with L + F . As we observed previously, travelling wave solutions (i.e. solutions of (1) satisfying xn (τ ) = xn−1 (τ − T /2)) appear as fixed points of S. This additional invariance implies that R1 = R S = S R is also a reversibility symmetry for Eq. (11). Reversible solutions under R1 correspond to solutions of (1)–(2) satisfying x−2n (−τ − T /2) = x2n+1 (τ ). The problem (11) is ill-posed as an initial value problem in D. Nevertheless, it is possible to construct bounded solutions for all t ∈ R. Using the method developed in [22], we are able to reduce (11) locally to a finite dimensional system of ordinary differential equations. The dimension of this reduced system depends on the bifurcation parameters γ and T (we shall fix T > 0 since Eq. (11) is even in T ). In the next section, we describe the spectrum of L in various parameter regions. 3. Spectral Problem The linear operator L is closed in H with domain D and has a compact resolvent. It follows that its spectrum consists of isolated eigenvalues σ with finite multiplicities. Let us compute the eigenvalues of L. Solving L U = σ U with U = (uˆ1 , uˆ2 , ξ1 , ξ2 , X1 , X2 )T leads to the equation A(uˆ1 , uˆ2 )T = 0,
where A=
σ 2 + T 2 (1 + 2γ ) −2T 2 γ cosh(σ/2) . −2T 2 γ cosh(σ/2) σ 2 + T 2 (1 + 2γ )
The dispersion relation detA = 0 reads N(σ, T , γ ) := (σ 2 + T 2 (1 + 2γ ))2 − 4(γ T 2 )2 cosh2 (σ/2) = 0.
(18)
The spectrum of L is then given by the roots of N (σ, T , γ ) = 0. Since L has real coefficients and due to the reversibility, the spectrum is invariant under the reflection on the real and the imaginary axis. We need basic properties of the spectrum in order to apply the reduction method [22]. As in reference [22], L is not bi-sectorial and the central part (σ = iq) of its spectrum is isolated from the hyperbolic part (σ = iq). More precisely, the following result can be obtained as in [22], p. 443.
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G. James, Y. Sire
Lemma 3.1. For all (γ , T ) ∈ R2+ , there exists p0 such that all eigenvalues σ = p + iq of L with p = 0 satisfy |p| ≥ p0 . For the central part of the spectrum (σ = iq), the dispersion relation reads (−q 2 + T 2 (1 + 2γ ))2 = 4(γ T 2 )2 cos2 (q/2).
(19)
In what follows we study the solutions of (19). Since (19) is even in q, we restrict ourselves to the case q ≥ 0. 3.1. Spectrum on the imaginary axis for γ T 2 < 4. The spectrum of L on the imaginary axis has a particularly simple structure for γ T 2 < 4. From the previous relation we deduce two cases: T 2 (1 + 2γ ) − q 2 = ±2γ T 2 cos(q/2).
(20)
Case + in (20). We consider the equation T 2 (1 + 2γ ) − q 2 = 2γ T 2 cos(q/2).
(21)
This equation can be written T 2 = q 2 − 4γ T 2 sin2 (q/4). We now consider
(T 2 , α2 )
as new parameters (recall α2 =
(22) γ T 2 ).
T 2 = fα2 (q) = q 2 − 4α2 sin2 (q/4).
Equation (22) reads (23)
If α2 < 4, fα2 : [0, +∞[→ R+ is a strictly increasing function of q and Eq. (23) yields q = fα−1 (T 2 ). This proves the existence of a pair of simple eigenvalues σ = ±ifα−1 (T 2 ) 2 2 2 ¯ for γ T < 4. The corresponding eigenvectors V , V read V = (1, 1, iq, iq, eiqv , eiqv )T . Note that R V = V¯ and S V = V . Case – in (20). We consider the equation T 2 (1 + 2γ ) − q 2 = −2γ T 2 cos(q/2).
(24)
In this case, we have T 2 = gα2 (q) = q 2 − 4α2 cos2 (q/4). If α2 < 4, gα2 : [0, +∞[→ R is a strictly increasing function of q and then q = gα−1 (T 2 ). 2 This proves the existence of another pair of simple eigenvalues σ = ±igα−1 (T 2 ) for 2 2 ¯ γ T < 4. The corresponding eigenvectors V , V read V = (−1, 1, −iq, iq, −eiqv , eiqv )T . We observe that R V = V¯ and S V = −V . Note that ifα−1 (T 2 ) = igα−1 (T 2 ) = i(2k + 1)π for T 2 (1 + 2γ ) = (2k + 1)2 π 2 2 2 (k ∈ N). In this case, the two pairs of eigenvalues collide, yielding a pair of double semi-simple eigenvalues (with eigenvectors having different symmetries). In what follows we extend the spectral study to the whole parameter space. In particular we shall consider the occurrence of double and triple purely imaginary eigenvalues.
Travelling Breathers in Klein-Gordon Chains
59
3.2. Double and triple eigenvalues on the imaginary axis. For having (at least) double ,γ ) = 0, i.e purely imaginary eigenvalues, we have to verify (19) and dN(iq,T dq 2q(−q 2 + T 2 (1 + 2γ )) = (γ T 2 )2 sin(q).
(25)
Moreover, iq is a triple eigenvalue when q satisfies (19),(25) and the following equation 2 ,T ) ( d N(iq,γ = 0): dq 2 −6q 2 + 2T 2 (1 + 2γ ) = (γ T 2 )2 cos(q).
(26)
The following lemma gives a description of the set of double and triple eigenvalues on the imaginary axis, as a function of (γ , T ) ∈ R2+ . These results are sketched in Fig. 1. Lemma 3.2. Consider the curve parametrized by (T (q), γ (q)) with q ∈ R+ and T , γ defined by the system (19)–(25). This curve (which we call a bifurcation curve) is given by: if q ∈ [4kπ, (2k + 1)2π ] (for an integer k ≥ 1), T 2 = q 2 − 4q tan(q/4), γ =
2q T 2 sin(q/2)
(27) (28)
,
Σ0
Σ9
Σ5
x Σ2
Σ3 x Σ4
*
x
Σ5
TP
ΤW
Σ1
Σ7
x
Σ8
Σ9
x
.. .. ....
γ
.. . .. . .. . . . . .
..
Σ15 Σ5
x
.... ...... ....
if q ∈ [(2k − 1)2π, 4kπ] (k ≥ 1),
Σ10 x
Σ5
Σ6
x
. . .
Σ11 x Σ12 x
Σ13
*
x
x
Σ7
Σ6
Σ6 Σ5
Σ12
TP Σ4
Σ6
2 Τ (1+2γ)=(2κ+1) 2 π 2
Σ7
Σ10 Σ8
Σ1
Σ1
Σ3
Σ4
κ=1
Σ1 Σ11 4π
.
x Double eigenvalue
Σ8 Σ10
Σ2
Simple eigenvalue
* Triple eigenvalue Σ11
2 γ Τ =4
Σ11 2π
Σ16 Σ14
Σ8
Σ8
Σ1
Σ0
Σ8
Σ13
Σ4 Σ4
Σ11 κ=0
Σ4
Σ6
Σ10
0
Σ15 x Σ16 x
Σ3
Σ4
Σ0
Σ14 x x
Σ12
κ=2
6π
κ=3
T
Fig. 1. Bifurcation curves and purely imaginary eigenvalues of L (upper half complex plane). “TP” (respectively “TW”) stands for the curves corresponding mainly to pulsating travelling wave (respectively travelling wave) bifurcations. The bold line corresponds to the subset
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G. James, Y. Sire
T 2 = q2 +
γ =−
4q , tan(q/4) 2q
T 2 sin(q/2)
.
(29)
(30)
The range of q is determined by the condition T 2 > 0. We denote by k the restriction of to the interval q ∈ [2kπ, 2(k + 1)π]. The curve lies in the parameter region where γ T 2 > 4. For (T , γ ) ∈ (except on a countable set of points ), the spectrum of L on the imaginary axis consists of a pair of double non-semi-simple eigenvalues ±i q and at least two distinct pairs of simple eigenvalues. The set of exceptional parameter values consists of the following types of points: • Cusps on correspond to the existence of a pair of triple eigenvalues ±i q¯ (Jordan block of index 3) satisfying tan(q/2) ¯ = q/2 ¯ and a pair of simple eigenvalues. • The point of tangent intersection between k and the curve T 2 (1 + 2γ ) = (2k + 1)2 π 2 leads to the existence of a pair of triple eigenvalues (with a two-dimensional eigenspace) and a pair of simple eigenvalues. • A point of transverse intersection between m and a curve T 2 (1+2γ ) = (2k+1)2 π 2 (k ∈ N) leads to the existence of two pairs of double eigenvalues (one being semisimple and the other non-semi-simple), and at least one pair of simple eigenvalues if m = k. • Double points on correspond to the existence of two pairs of double non semi-simple eigenvalues, and pairs of simple eigenvalues, depending on the parameter region. Proof. First, we divide (19) by (25) to obtain the following equation: T 2 (1 + 2γ ) = q 2 +
4q . tan(q/2)
(31)
Substituing the expression for T 2 (1 + 2γ ) in (25), we obtain γ =
2q . T 2 | sin(q/2)|
(32)
We have to consider two cases : sin(q/2) > 0 and sin(q/2) < 0. 2q Fixing γ = T 2 sin(q/2) in (31) yields T 2 = q 2 − 4q tan (q/4).
(33)
2q In the same way, fixing γ = − T 2 sin(q/2) in (31) leads to
T 2 = q2 +
4q . tan(q/4)
(34)
Furthermore, Eq. (32) shows that γ T 2 > 4. The spectrum of L on the imaginary axis as a function of γ , T is sketched in Fig. 1. The spectrum outside is obtained by continuity arguments.
Travelling Breathers in Klein-Gordon Chains
61
We note that for T 2 (1 + 2γ ) = (2k + 1)2 π 2 , k ∈ N, q∗ = (2k + 1)π is a solution of (20) for both cases + and −. Therefore, ±iq∗ = ±i(2k + 1)π is a pair of at least double eigenvalues. One can check that k has a tangent intersection with the curve T 2 (1 + 2γ ) = (2k + 1)2 π 2 at the point (T , γ ) = (T (q∗ ), γ (q∗ )). Moreover, Eq. (26) is satisfied at this point and consequently iq∗ is a triple eigenvalue of L (one can check that the associated eigenspace is two-dimensional). The existence of another pair of simple eigenvalues follows by a continuity argument. Moreover, one can show that k has only one other (transverse) intersection with the curve T 2 (1 + 2γ ) = (2k + 1)2 π 2 , at a point (T , γ ) = (T (q0 ), γ (q0 )) with q0 = q∗ . In this case one has two pairs of double eigenvalues (iq∗ being semi-simple and iq0 nonsemi-simple). Similar intersections between m (m = k) and T 2 (1+2γ ) = (2k +1)2 π 2 lead to extra pairs of simple eigenvalues. Finally, for q = (2k + 1)π Eqs. (31),(32) and (26) lead to tan(q/2) = (q/2).
(35)
In any fixed interval [2kπ, (2k + 1)π] (k ≥ 1) this equation has a unique solution q¯ (which determines γ , T uniquely). This solution corresponds to a triple eigenvalue i q¯ (and one has a Jordan block of index 3). Such triple eigenvalues appear as cusp points dγ of the bifurcation curve (( dT ¯
dq ) and ( dq ) vanish at q = q). Remark. Since our bifurcating solutions include the travelling waves found by Iooss and Kirchg¨assner [22], it is interesting to compare our bifurcation diagram with the one of reference [22]. More precisely, there exist travelling wave solutions of (1)–(2) (with p = 2) satisfying xn−1 (τ −
T ) = xn (τ ). 2
(36)
In order to establish a comparison of Lemma 3.2 with reference [22], we replace q by 2q in the parametrization of . This yields γ =
4q , T 2 | sin(q)|
(37)
and if q ∈ [2kπ, (2k + 1)π] T 2 = 4q 2 − 8q tan(q/2),
(38)
otherwise T 2 = 4q 2 +
8q . tan(q/2)
(39)
Now replacing T by 2T in (38) yields exactly the parametrization of the bifurcation curve given on p. 443 in [22]. Consequently, small amplitude solutions which bifurcate in the neighborhood of 2k include travelling wave solutions of reference [22]. These solutions can be combined with an additional mode corresponding to an extra pair of simple eigenvalues on the imaginary axis.
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G. James, Y. Sire
On the contrary, small amplitude solutions which bifurcate in the neighborhood of 2k+1 mainly consist (apart from spatially periodic travelling waves) of pulsating travelling waves not described in reference [22]. In what follows, we define as the subset of such that the central part of the spectrum is 0 = {±iq1 , ±iq2 , ±iq0 }, where ±iq0 is a pair of non semi-simple double eigenvalues and ±iq1 , ±iq2 two pairs of simple ones ( corresponds to the bold line in Fig. 1). One can check the following properties. Lemma 3.3. Fix (T , γ ) ∈ and let V0 ,V1 ,V2 be the eigenvectors associated to iq0 , iq1 , iq2 respectively. Denote by Vˆ0 the generalized eigenvector associated to iq0 . The eigenvectors can be chosen in the following way: V1 = (−1, 1, −iq1 , iq1 , −eiq1 v , eiq1 v )T , V2 = (1, 1, iq2 , iq2 , eiq2 v , eiq2 v )T , V0 = (, 1, iq0 , iq0 , eiq0 v , eiq0 v )T , Vˆ0 = (0, 0, , 1, veiq0 v , veiq0 v )T , where = −1 if q0 ∈ [(2k − 1)2π, 4kπ] and = 1 if q0 ∈ [4kπ, (2k + 1)2π ] (k ≥ 1). Moreover these eigenvectors satisfy RV0 = V0 , RV1 = V1 , RV2 = V2 , R Vˆ0 = −Vˆ0 , SV0 = V0 , SV1 = −V1 , SV2 = V2 , S Vˆ0 = Vˆ0 . 4. Optimal Regularity Problem and Reduction on a Center Manifold In this section we fix (T , γ ) ∈ , compute the spectral projection on the hyperbolic subspace (invariant subspace under L corresponding to the hyperbolic spectral part) and prove an optimal regularity result for the associated inhomogeneous linearized equation. This result is a crucial assumption for applying center manifold reduction theory [46]. Our proof closely follows the method given in [22]. We call P0 , P1 , P2 respectively the spectral projection on the 4-dimensional invariant subspace associated to ±iq0 , on the 2-dimensional subspace corresponding to ±iq1 , on the 2-dimensional subspace corresponding to ±iq2 . We also define P = P0 + P1 + P2 (spectral projection on the 8-dimensional central subspace) and use the notations Dh = (I − P )D, Hh = (I − P )H, Dc = P D, Uh = (I − P )U . The affine linearized system on Hh reads dUh = LUh + Fh (t), dt
(40)
where F (t) = (0, 0, f1 (t), f2 (t), 0, 0)T lies in the range of the nonlinear operator (13). We shall note Uh = (uh1 , uh2 , ξ1h , ξ2h , X1h (v), X2h (v))T . Our aim is to check the optimal regularity property of Eq. (40) (see [46], property (ii) p.127). This property can be stated as follows. We introduce the following Banach space, for a given Banach space Z and α ∈ R+ : Ejα (Z) = f ∈ C j (R, Z) f j = max sup e−α|t| |D k f (t)| < ∞ . (41) 0≤k≤j t∈R
We need to check that system (40) admits a unique solution Uh in E0α (Dh ) E1α (Hh ) for 0 ≤ α < α0 (for some α0 > 0), the operator Kh : E0α (R2 ) → E0α (Dh ), (f1 , f2 ) → Uh being bounded. As the linear operator L is not bi-sectorial, we do not have classical estimates on its resolvent and have to compute Uh explicitly.
Travelling Breathers in Klein-Gordon Chains
63
4.1. Computation of the spectral projection on the hyperbolic subspace. The spectral projection on the central subspace is defined by the Dunford integral 1 (σ I − L)−1 dC, (42) P = 2iπ C where C is a regular curve surrounding ±iq1 , ±iq2 , ±iq0 . The spectral projection on the hyperbolic subspace is Ph = I − P . We shall use the following result for computing Ph . (z) be a function of z ∈ C. Assume the function f (z) is entire Lemma 4.1. Let h(z) = fg(z) and the function g(z) admits a double pole at z = z0 . Then the residue of h at z = z0 is given by
Res(h, z0 ) =
2f (z0 )g (z0 ) − 23 f (z0 )g (z0 ) . g (z0 )2
(43)
In the following lemma, we compute the spectral projection on the hyperbolic subspace of a vector F lying in the range of the nonlinear operator (13). Lemma 4.2. Let F ∈ D be a vector of the type F = (0, 0, f1 , f2 , 0, 0)T . Then the projection of F on the hyperbolic subspace reads Fh = (0, 0, k3 f1 + k4 f2 , k5 f1 + k6 f2 , k7 (v)f1 + k8 (v)f2 , k9 (v)f1 + k10 (v)f2 )T , (44) where k3 , k4 , k5 , k6 ∈ R and k7 , k8 , k9 , k10 ∈ C ∞ ([−1/2, 1/2]) depend on γ , T . Proof. We first compute the resolvent of L. One has to solve (σ I − L)U = F , which yields the system ξ1 = σ u1 , ξ2 = σ u2 , (σ 2 − α1 )u1 − 2α2 cosh(σ/2)u2 = f1 , (σ 2 − α1 )u2 − 2α2 cosh(σ/2)u1 = f2 , (45) X1 (v) = u1 eσ v , X2 (v) = u2 eσ v , with U = (u1 , u2 , ξ1 , ξ2 , X1 (v), X2 (v))T . We have then 2 1 (σ − α1 )f1 + 2α2 cosh(σ/2)f2 u1 = . u2 N (σ, γ , T ) (σ 2 − α1 )f2 + 2α2 cosh(σ/2)f1
(46)
Now we compute the spectral projection P1 . Since σ = iq1 is a simple root of (18), one has Res(u1 , iq1 ) = Res(u2 , iq1 ) =
i(−(q12 + α1 )f1 + 2α2 cos(q1 /2)f2 ) 4q1 (q12 + α1 ) + 2α22 sin(q1 ) i(−(q12 + α1 )f2 + 2α2 cos(q1 /2)f1 ) 4q1 (q12 + α1 ) + 2α22 sin(q1 )
Denoting (P1 F )i the i th component of P1 F , we get consequently (P1 F )1 = Res(u1 , iq1 ) + Res(u1 , −iq1 ) = 0, (P1 F )2 = Res(u2 , iq1 ) + Res(u2 , −iq1 ) = 0.
, .
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G. James, Y. Sire
In the same spirit (P1 F )3 = (P1 F )4 = (P1 F )5 = (P1 F )6 =
−2q1 (−(q12 + α1 )f1 + 2α2 cos(q1 /2)f2 ) 4q1 (q12 + α1 ) + 2α22 sin(q1 ) −2q1 (−(q12 + α1 )f2 + 2α2 cos(q1 /2)f1 ) 4q1 (q12 + α1 ) + 2α22 sin(q1 )
,
,
−2 sin(q1 v)(−(q12 + α1 )f1 + 2α2 cos(q1 /2)f2 ) 4q1 (q12 + α1 ) + 2α22 sin(q1 ) −2 sin(q1 v)(−(q12 + α1 )f2 + 2α2 cos(q1 /2)f1 ) 4q1 (q12 + α1 ) + 2α22 sin(q1 )
, ,
which completes the computation of P1 F . The computations are identical for the spectral projection P2 associated to ±iq2 . For computing the spectral projection P0 associated to the double eigenvalues ±iq0 , we use formula (43). These computations lead to Eq. (44).
4.2. Resolution of the affine equation for bounded functions of t. We first solve (40) in the spaces Ejα with α = 0, i.e. we consider bounded functions of t (note that Ej0 (H) = j
Cb (H)). Fixing α = 0 will allow us to take the Fourier transform in time of the system in the tempered distributional space S (R). From (40), we directly deduce v h h X1 (t, v) = u1 (t + v) + (k7 (s)f1 (t + v − s) + k8 (s)f2 (t + v − s))ds (47) 0 t+v (k7 (t + v − s)f1 (s) + k8 (t + v − s)f2 (s))ds, (48) = uh1 (t + v) + t
X2h (t, v) = uh2 (t + v) + = uh2 (t + v) +
v
(k9 (s)f1 (t + v − s) + k10 (s)f2 (t + v − s))ds
(49)
0 t+v
(k9 (t + v − s)f1 (s) + k10 (t + v − s)f2 (s))ds (50)
t
(this expression comes from the two last equations of the affine linear system and from conditions X1 (0, t) = u1 (t),X2 (0, t) = u2 (t)). From the previous equations and the fact that (ki )i=7..10 and their derivatives are bounded functions of v, we deduce that X1h E 0 (C 1 [−1/2,1/2]) ≤ uh1 E 0 + C(f1 E 0 + f2 E 0 ),
(51)
X2h E 0 (C 1 [−1/2,1/2]) ≤ uh2 E 0 + C(f1 E 0 + f2 E 0 ).
(52)
0
0
1
1
0
0
0
0
We now have to estimate uh1 , uh2 , ξ1h , ξ2h . Taking the Fourier transform in time of the system (40) in the tempered distributional space S (R), we have (ik − L)Uˆh = Fˆh .
(53)
Travelling Breathers in Klein-Gordon Chains
65
We deduce ξˆ1h = ik uˆh1 , ξˆ2h = ik uˆh2 , Xˆ1h = eikv uˆh1 + fˆ1 Xˆ2h = eikv uˆh2 + fˆ1
v
e
ik(v−s)
k7 (s)ds + fˆ2
0 v
eik(v−s) k9 (s)ds + fˆ2
0
v
eik(v−s) k8 (s)ds,
0 v
eik(v−s) k10 (s)ds.
0
For uˆh1 , uˆh2 , we have −(k 2 + α1 )uˆh1 − 2α2 cos(k/2)uˆh2 = (Fˆh )3 + C1 (k)fˆ1 + C2 (k)fˆ2 , −(k 2 + α )uˆh − 2α cos(k/2)uˆh = (Fˆ ) + D (k)fˆ + D (k)fˆ , 1
2
2
h 4
1
1
1
2
(54)
2
where (Fˆh )3 = k3 fˆ1 + k4 fˆ2 , (Fˆh )4 = k5 fˆ1 + k6 fˆ2 , and Ci , Di are C ∞ functions of k, being O(1/|k|) as k → ±∞. Solving the system (54) leads to uˆh h1 N (ik, γ , T ) ˆ1 = , (55) h h 2 u2 where h1 = −(k 2 + α1 )[(k3 + C1 (k))fˆ1 + (k4 + C2 (k))fˆ2 ] + 2α2 cos(k/2)[(k5 + D1 (k))fˆ1 +(k6 + D2 (k))fˆ2 ], h2 = 2α2 cos(k/2)[(k3 + C1 (k))fˆ1 + (k4 + C2 (k))fˆ2 ] − (k 2 + α1 )[(k5 + D1 (k))fˆ1 +(k6 + D2 (k))fˆ2 ]. Equation (55) can be written N (ik, γ , T )
uˆh1 + Hˆ 1 fˆ1 + Hˆ 2 fˆ2 ˆ fˆ + G ˆ fˆ uˆh + G 2
1 1
2 2
=
0 . 0
(56)
As the operator (ik − Lh )−1 is analytic in a strip around the real axis, we deduce that ˆ 1, G ˆ 2 are analytic functions in this strip. Moreover, Hˆ 1 , Hˆ 2 , G ˆ 1, G ˆ 2 are O( 12 ) Hˆ 1 , Hˆ 2 , G k
as k → ±∞ due to the fact that N (ik, γ , T ) = O(k 4 ) and h1 , h2 are O(k 2 ) as k → ±∞. Since N(iqj , γ , T ) = 0, N (iq0 , γ , T ) = 0 and N (iq1 , γ , T ), N (iq2 , γ , T ), N (iq0 , γ , T ) do not vanish, Eq. (56) yields uˆh1 + Hˆ 1 fˆ1 + Hˆ 2 fˆ2 = a1+ δiq1 + a1− δ−iq1 + a2+ δiq2 + a2− δ−iq2
+ b0− δ−iq , +a0+ δiq0 + a0− δ−iq0 + b0+ δiq 0 0
(57)
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G. James, Y. Sire
ˆ 1 fˆ1 + G ˆ 2 fˆ2 = c+ δiq1 + c− δ−iq1 + c+ δiq2 + c− δ−iq2 uˆh2 + G 1 1 2 2
+c0+ δiq0 + c0− δ−iq0 + d0+ δiq + d0− δ−iq . 0 0
(58)
ˆ i belong to L2 (R). Therefore, Furthermore, k → (1+|k|2 )1/2 Hˆ i and k → (1+|k|2 )1/2 G using the inverse Fourier Transform and Lemma 3, p.448 of [22], there exist Gi , Hi ∈ ˆ i , Hˆ i Hδ1 (R) (i.e eδ|t| Hi ∈ H 1 (R), eδ|t| Gi ∈ H 1 (R), δ > 0 small enough) such that G are the unique Fourier transforms of Gi , Hi . We have the following estimates dH1 dH1 (t − s)f1 (s)ds| ∗ f1 C 0 = sup | b dt R dt t∈R ≤ C(δ)f1 C 0 H1 H 1 (R) . (59) b
δ
dG1 dG2 2 The same estimate is valid for dH dt ∗ f2 , dt ∗ f1 , dt ∗ f2 . Now we make the solution of (40) explicit. We set U˜ h = (u˜ h1 , u˜ h2 , ξ˜1h , ξ˜2h , X˜ 1h , X˜ 2h )T and
u˜ h1 = −H1 ∗ f1 − H2 ∗ f2 , u˜ h2 = −G1 ∗ f1 − G2 ∗ f2 , d u˜ h1 , dt d u˜ h2 ξ˜2h = , dt
ξ˜1h =
(60)
X˜ 1h (t, v) = u˜ h1 (t + v) + X˜ 2h (t, v) = u˜ h2 (t + v) +
v
0 v
(k7 (s)f1 (t + v − s) + k8 (s)f2 (t + v − s))ds, (k9 (s)f1 (t + v − s) + k10 (s)f2 (t + v − s))ds.
0
By construction, u˜ h satisfies (40) and P Uˆ˜ h = 0 (hence P U˜ h = 0) for (f1 , f2 ) ∈ E0α (R2 ) with α < 0 (fˆi are analytic functions in a strip around the real axis). Since the computations are formally the same for α = 0, we have P Uˆ˜ h = 0 for α = 0, hence P U˜ h = 0 for α = 0. Moreover, we have U˜ h C 0 (Dh ) C 1 (Hh ) ≤ C(f1 C 0 (R) + f2 C 0 (R) ) b
b
b
b
(61)
due to estimates (51), (52), (59) (with analogous estimates on H2 , Gi ). For α = 0, we obtain uh1 , uh2 by adding to u˜ h1 , u˜ h2 the inverse Fourier transforms of Dirac measures, i.e. uh1 = u˜ h1 + a1+ eiq1 t + a1− e−iq1 t + a2+ eiq2 t + a2− e−iq2 t +(a0+ + itb0+ )eiq0 t + (a0− − itb0− )e−iq0 t ,
(62)
uh2 = u˜ h2 + c1+ eiq1 t + c1− e−iq1 t + c2+ eiq2 t + c2− e−iq2 t +(c0+ + itd0+ )eiq0 t + (c0− − itd0− )e−iq0 t .
(63)
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Since P U˜ h = 0, we have P Uh = 0 if and only if a1± = a2± = c1± = c2± = b0± = a0± = d0± = 0.
(64)
It follows that Uh = U˜ h . Finally, we have proved the following Lemma 4.3. Assume F = (0, 0, f1 , f2 , 0, 0)T and f1 , f2 ∈ Cb0 (R). Then the affine linear system (40) has a unique bounded solution Uh ∈ Cb0 (Dh ) Cb1 (Hh ) and the operator Kh : Cb0 (R2 ) → Cb0 (Dh ), (f1 , f2 ) → Uh is bounded. Remark. The first and second components of (44) vanish due to our choice of variables (u1 , u2 ) in (7). This would not be the case using (x1 , x2 ) and the proof of optimal regˆ i are only O(1/|k|) as ularity results would require additional work (in this case Hˆ i ,G k → ±∞). 4.3. Affine equation in exponentially weighted spaces. The problem now is to extend Lemma 4.3 to the case (f1 , f2 ) ∈ E0α (R2 ), with α > 0 sufficiently close to 0. This has been done in [22] by constructing a suitable distribution space, but the following lemma gives an alternative proof (see [34]). Lemma 4.4. Consider Banach spaces D,Y and X such that: D → Y → X. Let L be a closed linear operator in X, of domain D, such that the equation dU = LU + f dt
(65)
U E0α (D) ≤ C(α)f E0α (Y) .
(66)
admits for any fixed f ∈ Cb0 (Y) a unique solution U = Kf in Cb0 (D) Cb1 (X), with in addition K ∈ L(Cb0 (Y), Cb0 (D)). Then there exists α0 > 0 such that if 0 ≤ α < α0 , for all f ∈ E0α (Y) the system (65) admits a unique solution in E0α (D) E1α (X) with
f (t) U (t) Proof. Let f ∈ E0α (Y). We set: f˜(t) = cosh(αt) ∈ Cb0 (Y) and U˜ (t) = cosh(αt) . The α α 0 1 property U ∈ E0 (D) E1 (X) is equivalent to U˜ ∈ Cb (D) Cb (X). Furthermore, we have d U˜ = LU˜ + f˜ − α tanh(αt)U˜ . dt This equation is equivalent to
U˜ + αK(tanh(αt)U˜ ) = K f˜.
(67)
Equation (67) can be written (I + αT )U˜ = K f˜, where T U˜ = K(tanh(αt)U˜ ). We have then T ∈ L(Cb0 (D)) and T ≤ K. If 0 ≤ 1 1 α < K , I + αT is invertible in Cb0 (D) and we have (I + αT )−1 ≤ 1−αK . Therefore, (67) is equivalent to U˜ = (I + αT )−1 K f˜ ∈ Cb0 (D)
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and U˜ C 0 (D) ≤ b
K ˜ 1−αK f Cb0 (Y) .
U E0α (D) ≤ U˜ C 0 (D) ≤ b
This ends the proof.
Then, we have U = cosh(αt)U˜ ∈ E0α (D) and
K K f˜C 0 (Y) = 2 f E0α (Y) . b 1 − αK 1 − αK
(68)
Applying this result to our problem yields the following. Proposition 4.5. There exists α0 > 0 such that for all F = (0, 0, f1 , f2 , 0, 0)T with f1 , f2 ∈ E0α (R) and α ∈ [0, α0 ], the affine linear system (40) has a unique solu tion Uh ∈ E0α (Dh ) E1α (Hh ). Moreover, the operator Kh : E0α (R2 ) → E0α (Dh ), (f1 , f2 ) → Uh is bounded (uniformly in α ∈ [0, α0 ]). 4.4. Center manifold reduction. The above analysis shows that the assumptions of Theorem 3 of reference [46] (p. 133) are satisfied. Hence the reduction on a center manifold is possible and we have the following result. Theorem 4.1. Fix (T0 , γ0 ) ∈ and k ≥ 1. There exists a neighborhood U × V of (0, γ0 , T0 ) in D × R2 and a map ψ ∈ Cbk (Dc × R2 , Dh ) such that the following properties hold for all (γ , T ) ∈ V (with ψ(0, γ , T ) = 0, Dψ(0, γ0 , T0 ) = 0). • If U : R → D solves (11) and U (t) ∈ U ∀t ∈ R then Uh (t) = ψ(Uc (t), γ , T ) for all t ∈ R and Uc is a solution of dUc (69) = LUc + P F (Uc + ψ(Uc , γ , T )). dt • If Uc : R → Dc is a solution of (69) with Uc ∈ Uc = P U ∀t ∈ R, then U = Uc + ψ(Uc , γ , T ) is a solution of (11). • The map ψ(., γ , T ) commutes with R and S. Moreover, the reduced system (69) is reversible under R and equivariant under S. 5. Study of the Reduced Equation According to normal form theory (see e.g. [21]), one can perform a polynomial change of variables Uc = U˜ c + P˜γ ,T (U˜ c ) close to the identity which simplifies the reduced Eq. (69) and preserves its symmetries. In this section, we compute this normal form at order 3 and give an explicit expression of a particular coefficient, which sign is essential for the bifurcation of small amplitude homoclinic orbits. 5.1. Normal form computation. The linear operator L restricted to the eight-dimensional subspace Dc (denoted as Lc ) has the following structure in the basis (V0 , Vˆ0 , V1 , V2 , V¯0 , V¯ˆ 0 , V¯1 , V¯2 ): iq0 1 0 0 0 0 0 0 0 iq0 0 0 0 0 0 0 0 0 iq1 0 0 0 0 0 0 0 0 0 0 0 iq2 0 . Lc = 0 0 0 0 0 0 −iq0 1 0 0 0 0 0 −iq 0 0 0 0 0 0 0 0 0 −iq1 0 0 0 0 0 0 0 0 −iq2
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Moreover, the reversibility symmetry R and the symmetry S have the following structure. One has 0 0 001 0 00 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 R= . 1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 010 0 00 Moreover, if (T0 , γ0 ) ∈ 2k we have S = diag(1, 1, −1, 1, 1, 1, −1, 1) and (T0 , γ0 ) ∈ 2k+1 yields S = diag(−1, −1, −1, 1, −1, −1, −1, 1). Consequently, our reduced equation has many similarities with the one considered in [22], Sect. 6 (case of a (iq0 )2 (iq2 ) resonance). The only differences are an extra pair of simple purely imaginary eigenvalues ±iq1 for the linearized operator, and the additional symmetry S. More precisely, the truncated normal form considered in [22] has a symmetry similar to S (which follows in fact from a phase invariance), but this symmetry is broken for the full system. In our case, Theorem 4.1 ensures that the full reduced system is equivariant under S. It follows that the normal form has a structure similar to the one obtained in [22]. To compute the normal form, we exclude points of which are close to points where sq0 + rq1 + r q2 = 0 for s, r, r ∈ Z and 0 < |s| + |r| + |r | ≤ 4 (such values correspond to strong resonances), and denote this new set as 0 . The normal form computation is very similar to [22] (Sect. 6 and Appendix 2), to which we refer for details. In what follows we set U˜ c = AV0 +B Vˆ0 +CV1 +DV2 +A V 0 +B Vˆ0 +C V 1 +D V 2 . The normal form of (69) at order 3 is given in the following lemma. Lemma 5.1. The normal form of (69) at order 3 reads dA = iq0 A + B + iAP(u1 , u2 , u3 , u4 ) dt +O((|A| + |B| + |C| + |D|)4 ), dB = iq0 B + iBP(u1 , u2 , u3 , u4 ) + AS(u1 , u2 , u3 , u4 ) dt +O((|A| + |B| + |C| + |D|)4 ), dC = iq1 C + iCQ(u1 , u2 , u3 , u4 ) + O((|A| + |B| + |C| + |D|)4 ), dt dD = iq2 D + iDT (u1 , u2 , u3 , u4 ) + O((|A| + |B| + |C| + |D|)4 ), dt where ¯ u2 = C C, ¯ u3 = D D, ¯ u4 = i(AB¯ − AB) ¯ u1 = AA,
(70)
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and P, S, Q, T are polynomials with smooth parameter dependent real coefficients, for (T , γ ) in the neighborhood of 0 .We have P(u1 , u2 , u3 , u4 ) = p1 (γ , T ) + p2 u1 + p3 u2 + p4 u3 + p5 u4 , S(u1 , u2 , u3 , u4 ) = s1 (γ , T ) + s2 u1 + s3 u2 + s4 u3 + s5 u4 , Q(u1 , u2 , u3 , u4 ) = q˜1 (γ , T ) + q˜2 u1 + q˜3 u2 + q˜4 u3 + q˜5 u4 , T (u1 , u2 , u3 , u4 ) = t1 (γ , T ) + t2 u1 + t3 u2 + t4 u3 + t5 u4 ,
(71)
where p1 , s1 , q˜1 , t1 vanish on 0 . The truncated normal form (obtained by neglecting terms of orders 4 and higher) is integrable with the following first integrals: |A|2 ¯ ¯ |B|2 − S(x, |C|2 , |D|2 , i(AB¯ − AB))dx, |C|2 , |D|2 . (72) AB¯ − AB, 0
Note that if one fixes |C| = 0 and |D| = 0, the truncated normal form yields the classical 1:1 resonance [24]. In what follows we describe some solutions of the truncated normal form. We shall concentrate on the description of homoclinic solutions to the equilibrium 0, to a periodic or a quasi-periodic orbit, which may exist when Lc has 4 eigenvalues with nonzero real parts (perturbation of ±iq0 ). The existence of these homoclinic orbits is linked to the sign of the coefficient s2 in the polynomial S. The following section is devoted to its computation. 5.2. Computation of the coefficient s2 . We choose (T0 , γ0 ) ∈ 0 . Equation (11) can be expanded as dU = L0 U + (γ − γ0 )L(1) U + (T − T0 )L(2) U + M2 (U, U ) + M3 (U, U, U ) + ...., dt (73) where L0 is the linear operator for (T0 , γ0 ) ∈ 0 and L(i) are linear operators. Moreover, Mj is a j −linear symmetric map satisfying M2 (U, U ) = aT02 (0, 0, u21 , u22 , 0, 0)T ,
(74)
M3 (U, U, U ) = bT02 (0, 0, u31 , u32 , 0, 0)T .
(75)
Using the Taylor expansion of the center manifold at (0, γ0 , T0 ) we find U = AV0 + B Vˆ0 + CV1 + DV2 + A V 0 + B Vˆ0 + C V 1 + D V 2 + (m,n) (γ − γ0 )m (T − T0 )n Ar0 B rˆ0 C r1 D r2 A¯ s0 B¯ sˆ0 C¯ s1 D¯ s2 φr rˆ r r s sˆ s s . 0 0 1 2 0 0 1 2
(76)
Using this expression and the normal form in Eq. (73), we find by identification at orders A2 , |A|2 , A|A|2 (we omit the index (m, n) = (0, 0) in the notations)
ip2 V0 + s2 Vˆ0 + (iq0 I − L)φ20001000
(2iq0 I − L)φ20000000 = M2 (V0 , V0 ), −Lφ10001000 = 2M2 (V0 , V¯0 ), = 2M2 (V¯0 , φ20000000 ) + 2M2 (V0 , φ10001000 ) + 3M3 (V0 , V0 , V¯0 ).
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The first two equations have a unique solution given by expressions (45), (46). The last equation yields the following compatibility condition (expression (45) reduces the problem to a two-dimensional system) (2 −
4a 2 T02 q0 ). )s2 = T02 (6b + 8a 2 − tan(q0 /2) 2γ0 T02 cos(q0 ) − T02 (1 + 2γ0 ) + 4q02
(77)
The other coefficients in (71) could be computed by identification in a similar way. 5.3. Description of small amplitude solutions for the normal form system. This section describes some reversible homoclinic solutions of the truncated normal form given in Lemma 5.1. The problem of their persistence for the full system is discussed in different cases. We choose (γ , T ) ≈ (γ0 , T0 ) ((T0 , γ0 ) ∈ 0 ), in such a way that the linearized operator L has four symmetric eigenvalues close to ±iq0 and having non-zero real parts (s1 (γ , T ) > 0 in (71)). We shall distinguish S-invariant and non-S-invariant solutions, where S is the permutational symmetry (17). We recall that S-invariant solutions correspond to travelling waves. 5.3.1. Solutions bifurcating at (T0 , γ0 ) ∈ 2k . • S-invariant homoclinic solutions and persistence problems We consider the normal form system (70) restricted to the invariant subspace Fix(S). In this case we have C = 0 and recover the (iq0 )2 (iq2 ) resonance case as in [22]. The subspace Fix(S) contains in particular the stable and unstable manifolds of 0. Provided s2 (γ0 , T0 ) < 0 and (γ , T ) ≈ (γ0 , T0 ), the truncated normal form system admits homoclinic orbits to 0 with D = 0. In addition there exist homoclinic solutions to small periodic orbits with D = 0. These solutions are given by (α ≈ 0), A(t) = r0 (t)ei(q0 t+ψ(t)+θ) , B(t) = r1 (t)ei(q0 t+ψ(t)+θ) , D(t) = αei(q2 t+ϕ2 (t)+θ2 ) , where 2(s1 + s4 α 2 ) 1/2 ) (cosh(t (s1 + s4 α 2 )1/2 ))−1 , −s2 dr0 r1 (t) = (t), dt p2 ψ(t) = (p1 + p4 α 2 )t + 2 (s1 + s4 α 2 )1/2 tanh(t (s1 + s4 α 2 )1/2 ), s2 t 2 ϕ2 (t) = (t1 + t4 α ) t + t2 r02 (τ ) dτ , r0 (t) = (
0
and θ, θ2 ∈ R. These orbits are reversible under R if one chooses θ and θ2 equal to 0 or π . In this case, the problem of their persistence for the full vector field (with additional nonresonance conditions on the eigenvalues) has been treated by Lombardi in [28]. Reversible homoclinic solutions to periodic orbits persist above a critical tail size α = αc , which is exponentially small with respect to |A(0)| (size of “central” oscillations). This yields exact travelling wave solutions of the Klein-Gordon system [22], which converge towards
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periodic waves at infinity and have a larger amplitude at the center of the chain. On the contrary, reversible homoclinic orbits to 0 should not persist generically for the full normal form (70) when higher order terms are taken into account [28]. In what follows we explain this statement in more detail and give a brief account of persistence and nonpersistence results obtained in [28]. Consider the normal form (70) restricted to the invariant subspace C = 0. We fix (T0 , γ0 ) ∈ 0 ∩ 2k , with additional nonresonance conditions on the eigenvalues (see [28], p. 359) which are generically realized. We assume s2 (γ0 , T0 ) < 0 and s1 (γ , T ) > 0. For simplicity we fix γ = γ0 and let T ≈ T0 vary. In the linearized system, 4 hyperbolic eigenvalues have small real parts ±ν = O(|T −T0 |1/2 ) with ν > 0 (we shall use ν instead of T − T0 as a small parameter), O(1) imaginary parts ±iω0 (ν) (ω0 (0) = q0 ), and there is in addition one pair of O(1) purely imaginary eigenvalues ±iω2 (ν) (ω2 (0) = q2 ). Using the following scaling (see [28], p. 364) ˜ ˜ ˜ ˜ C(t) = ν 3/2 C(νt), D(t) = ν 3/2 D(νt) A(t) = σ ν A(νt), B(t) = σ ν 2 B(νt), with σ = (−2/s2 )1/2 , the normal form (70) can be written dY = N (Y, ν) + R(Y, ν), dt
(78)
¯˜ B, ¯˜ C, ¯˜ D, ¯˜ T . The linearized system has the eigenvalues ±1± ˜ B, ˜ A, ˜ C, ˜ D) where Y = (A, iω0 /ν and ±iω2 /ν (a slow hyperbolic part coincides with fast oscillatory parts in this system). Moreover N is a cubic polynomial in Y , R contains higher order terms in Y and is O(ν) as ν → 0. The truncated system (with R = 0) has explicit reversible solutions ±h homoclinic to 0, being O(1) as ν → 0 thanks to the scaling (the unscaled solutions have been given above). The rescaled solution h has simple poles z = ±iπ/2 in the complex plane (one has r0 (t) = 1/ cosh t in the above notations). We start with some comments on the generic nonpersistence of reversible homoclinic orbits to 0 [28]. Setting Y = h + y, where the perturbation y is assumed reversible under R and homoclinic to 0, (78) can be rewritten in the form dy − DY N (h(t), ν) y = f (y, h(t), ν). dt
(79)
Applying the Fredholm alternative, one obtains a compatibility condition (linked to the eigenvalues ±iω2 /ν and reversibility) having the form +∞ y ∗ (t), f (y(t), h(t), ν) dt = 0, (80) 0
where the dual vector
y∗
reads
y ∗ (t) = (0, 0, −ieiψr (t) , 0, 0, ie−iψr (t) ), and ψr has the form ψr (t) = ω2 t/ν + ν n(ν) tanh t. This yields a condition of the type +∞ I (ν) = Im e−iω2 t/ν g(y(t), h(t), t, ν) dt = 0, (81) 0
consisting of a bi-oscillatory integral in which the approximate homoclinic solution h also rotates at the high frequency ω0 /ν.
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The usual way to check if I (ν) vanishes is to split the integral in two parts I (ν) = Me (ν) + J (ν), where the Melnikov function +∞ e−iω2 t/ν g(0, h(t), t, ν) dt (82) Me (ν) = Im 0
depends on the explicitly known function h and is usually expected (at least in classical perturbation theory) to be the leading part of I (ν). One finds as ν → 0 (see [28], p. 397) Me (ν) = ν −3/2 e−c/ν (1 + O(ν)) (c > 0), hence Me (ν) is exponentially small. However, Me (ν) is not the leading part of I (ν) in our case. Indeed, fine estimation techniques [28] yield I (ν) = ν −3/2 e−c/ν ( + O(ν 1/4 ))
(83)
with = 1 in general. The reason is that h is the leading part of Y on R, but not near the poles of Y (close to z = ±iπ/2), and the leading part of Y near the poles is precisely the relevant part for computing I (ν). More precisely, the coefficient in (83) is given by a complex integral, involving a (not explicitly known) solution on the stable manifold of Y = 0, extended in the complex plane and approximated near the poles ±iπ/2 at leading order (see [29, 28]). As a consequence, analytical computations of seem very difficult but numerical ones might be achieved. Moreover, an additional difficulty for obtaining estimate (83) has to be pointed out. Since center manifolds are not analytic in general (not even C ∞ ), one cannot work with the (a priori) non-analytic reduced Eq. (70). In order to preserve analyticity, one works directly with the evolution problem (11), splitted into an infinite-dimensional hyperbolic part coupled with the normal form (70), whose principal part remains unchanged (see [28], p. 331). The same techniques as in the finitedimensional case apply, because Lemma 4.3 and Eq. (60) give the necessary optimal regularity properties for the hyperbolic part of the linearized system (see [28], Sect. 8). According to expression (83), if = 0 (which should be satisfied except for exceptional choices of (T0 , γ0 ) and V ) and T − T0 is sufficiently small, reversible homoclinic orbit to 0 close to ±h do not exist. Consequently, reversible homoclinic orbits to 0 should not persist generically for the full normal form. This result needs several comments. Firstly, it might happen that = 0 for isolated values of (T0 , γ0 ) ∈ 2k . In that case, one might expect the existence of a curve I (T , γ ) = 0 in the parameter plane (with (T , γ ) ≈ (T0 , γ0 )) on which the compatibility condition (81) is satisfied and reversible homoclinic orbits to 0 exist. However this situation is non-generic in the parameter plane. Moreover, the above analysis only concerns reversible homoclinic orbits, and nonreversible homoclinic orbits to 0 might exist. In addition, homoclinic solutions are searched in a small neighborhood of h in L∞ (R), and reversible homoclinic orbits with several loops (which do not satisfy this criteria) might exist as it is mentioned in [28]. Consequently, = 0 only implies the nonexistence of homoclinic orbits to 0 of a certain type when ν is small enough. We end with some precisions about persistence of reversible solutions homoclinic to periodic orbits. One can show [28] that for ν small enough and α in an interval of the type α ∈ (K1 e−a/ν , K2 ),
(84)
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(a > 0), Eq. (78) admits reversible solutions of the form Yα,ν (t) = y(t) + h(t) + Xα,ν (t + ϕ tanh (λ t)),
(85)
where y is homoclinic to 0 and Xα,ν is a reversible time-periodic solution of (78) with amplitude α. The frequency of Xα,ν is close to ω2 /ν and its principal part (in the unscaled form) has been given above (case A = B = C = 0 in the truncated normal form (70)). Very roughly speaking, looking for a solution of the form (85) yields a compatibility condition of the type +∞ α sin ϕ = Im e−iω2 t/ν G(y(t), h(t), t, α, ν, ϕ) dt, (86) 0
which holds for a suitable choice of the phase ϕ = ϕ(α, ν) provided (84) is satisfied, due to the exponential smallness of the right side of (86) (see [28], Sect. 9.3 for more details). • Non S-invariant solutions Provided s2 (γ0 , T0 ) < 0 and (γ , T ) ≈ (γ0 , T0 ), the truncated normal form admits homoclinic solutions to small quasi-periodic orbits, which are not invariant under S due to the additional component C(t). These solutions are given by (α, β ≈ 0, β = 0) A(t) = r0 (t)ei(q0 t+ψ(t)+θ) , B(t) = r1 (t)ei(q0 t+ψ(t)+θ) , C(t) = βei(q1 t+ϕ1 (t)+θ1 ) , D(t) = αei(q2 t+ϕ2 (t)+θ2 ) ,
(87)
where (˜s = s1 + s4 α 2 + s3 β 2 ) 2˜s 1/2 ) (cosh(t s˜ 1/2 ))−1 , −s2 dr0 r1 (t) = (t), dt p2 ψ(t) = (p1 + p4 α 2 + p3 β 2 )t + 2 s˜ 1/2 tanh(t s˜ 1/2 ), s2 t ϕ1 (t) = (q˜1 + q˜4 α 2 + q˜3 β 2 ) t + q˜2 r02 (τ ) dτ , 0 t 2 2 ϕ2 (t) = (t1 + t4 α + t3 β ) t + t2 r02 (τ ) dτ , r0 (t) = (
0
and θ, θ1 , θ2 ∈ R. This family of solutions does not include homoclinic orbits to 0, since the latter are S-invariant. These orbits are reversible under R if one chooses θ, θ1 and θ2 equal to 0 or π, and reversible under R1 = R S if one chooses θ1 = ±π/2 and θ, θ2 equal to 0 or π . The persistence of these orbits for the full vector field is still an open problem. In the reversible cases this may be analyzed using techniques developed by Lombardi [28] for the (iq0 )2 iq2 resonance (see the above paragraph on S-invariant solutions), but the extra pair of eigenvalues ±iq1 makes the problem more difficult. For β |A(0)|, solutions (87) of the truncated normal form correspond to approximate solutions of the Klein-Gordon system, consisting of a travelling wave superposed on a small oscillatory mode (mainly visible at the tail).
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5.3.2. Solutions bifurcating at (T0 , γ0 ) ∈ 2k+1 . • S-invariant solutions We consider solutions of the truncated normal form (70) on the invariant subspace Fix(S). These solutions satisfy A = B = C = 0. They are periodic, given by ∗ D(t) = αeiω t+θ2 with ω∗ = q2 + t1 + t4 α 2 . Their persistence for the full vector field (restricted to Fix(S)) follows from the Devaney-Lyapunov theorem. These solutions correspond to spatially periodic travelling waves of the Klein-Gordon system, which have been obtained in [22]. • Non S-invariant solutions For s2 (γ0 , T0 ) < 0 and (γ , T ) ≈ (γ0 , T0 ), the truncated normal form admits homoclinic solutions to small quasi-periodic orbits, given by Eq. (87). Their persistence for the full vector field is still an open problem. For reversible solutions this problem may be treated using the techniques developed by Lombardi [28], but in the present case an extra pair of purely imaginary eigenvalues makes the problem more difficult. • The existence of homoclinic orbits to 0 reversible under R would be only possible with two compatibility conditions satisfied. The situation is similar to Sect. 5.3.1 ((iq0 )2 iq2 resonance for S-invariant solutions), except one obtains in the present case one compatibility condition for each pair of simple purely imaginary eigenvalues. Here the linearized system has 4 hyperbolic eigenvalues with small real parts ±ν (we shall use ν as a small parameter) and O(1) imaginary parts ±iω0 (ν) (ω0 (0) = q0 ), and there are in addition two pairs of O(1) purely imaginary eigenvalues ±iω1 (ν), ±iω2 (ν) (ωj (0) = qj ). Using the same notations as in Sect. 5.3.1, compatibility conditions take the form of oscillatory integrals +∞ e−iω1 t/ν g1 (y(t), h(t), t, ν) dt = 0, (88) I1 (ν) = Im 0
+∞
I2 (ν) = Im
e−iω2 t/ν g2 (y(t), h(t), t, ν) dt = 0.
(89)
0
As in Sect. 5.3.1, h(t)+y(t) denotes a reversible homoclinic orbit to 0 of the rescaled reduced equation. Its principal part h(t) is explicit and given (in the unscaled form) by (87) (with C = D = 0 and θ equal to 0 or π). The existence of homoclinic orbits to 0 reversible under R1 = R S would imply two compatibility conditions similar to (88)–(89) (one has θ = ±π/2 in (87) and one takes the real part of the integral in (88)). Instead of homoclinic orbits to 0, we conjecture the persistence of reversible homoclinic orbits to exponentially small 2−dimensional tori, originating from the two additional pairs of simple imaginary eigenvalues. As we shall see, solutions (87) of the truncated normal form correspond to approximate solutions of the Klein-Gordon system, consisting of a pulsating travelling wave with oscillations of size |A(0)| at the center. 5.3.3. Persistence result in a particular case. We consider the case when the potential V in (1) is even (case a = 0 in (10)). Due to the additional invariance xn → −xn of (1), Eq. (11) is also invariant under −S. Fixed points of −S correspond to solutions of (1) satisfying T xn+1 (τ ) = −xn (τ − ). 2
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In this case we have xn (τ ) = (−1)n+1 x1 (τ − advance-delay differential equation
(n−1)T 2
) and x1 satisfies a simpler scalar
d 2 x1 (τ ) + V (x1 (τ )) = −γ (x1 (τ + T /2) + 2x1 (τ ) + x1 (τ − T /2)). dτ 2
(90)
For (T0 , γ0 ) ∈ 2k+1 , the symmetry −S has the following structure on the central subspace −S = diag(1, 1, 1, −1, 1, 1, 1, −1). We consider the normal form (70) on the invariant subspace Fix(−S), which corresponds to fixing D = 0. In particular, the stable and unstable manifolds of 0 are included in Fix(−S). By considering the flow on Fix(−S), we recover the (iq0 )2 (iq1 ) resonance case treated in [28] and summarized in Sect. 5.3.1. Under non-resonance assumptions ( qq01 = pq
for p + q ≤ 5, qq01 ∈ N2 and qq01 ∈ N), reversible homoclinic solutions to periodic orbits given by (87) (with D = 0) persist for the full vector field above a critical tail size β = βc , which is exponentially small with respect to |A(0)| (size of “central” oscillations). These solutions are either reversible under R (for θ , θ1 equal to 0 or π in (87)) or R1 = R S = −R (for θ, θ1 equal to ±π/2). As we shall see in Sect. 6, these orbits yield exact travelling breather solutions of the Klein-Gordon system, superposed on an exponentially small oscillatory tail. Homoclinic orbits to 0 reversible under R do not persist for the full normal form (70) if the compatibility condition (88) (corresponding to the pair of eigenvalues ±iq1 ) is not satisfied [28]. A similar condition holds for reversible solutions under −R. Note that the compatibility condition (89) is automatically satisfied by fixing a = 0 in V , thanks to the symmetry −S of (70). Indeed, the stable manifold of 0 has no D-component and the D-component of the full normal form (70) vanishes for D = 0, which implies the vanishing of g2 in (89). As in Sect. 5.3.1 for S-invariant solutions, there might be a discrete collection of curves I1 (T , γ ) = 0 in the parameter plane (with (T , γ ) ≈ 2k+1 ) on which the relevant compatibility condition would be satisfied and reversible homoclinic orbits to 0 would exist. In the next section, we study the sign of the crucial normal form coefficient s2 for (γ0 , T0 ) ∈ 0 (homoclinic orbits are found for s2 < 0). 5.4. Sign of the bifurcation coefficient s2 . In the following, we determine the sign of the coefficient s2 (γ0 , T0 ) as a function of the parameters (T0 , γ0 ) ∈ k and parameters a,b in the potential (see (10)). We recall that the homoclinic solutions (87) exist for s2 (γ0 , T0 ) < 0 and (T0 , γ0 ) ∈ 0 . 5.4.1. Case of an even potential (a = 0) For a = 0 we have s2 =
3T02 b 1−
q0 2 tan(q0 /2)
.
Let us define Z(q0 ) = 1 −
q0 . 2 tan(q0 /2)
(91)
Travelling Breathers in Klein-Gordon Chains
77
We have then sign(s2 ) = sign(b)sign(Z(q0 )).
(92)
By Lemma 3.2, one has Z = 0 at cusp points of the bifurcation curve (these points have been removed from the parameter set 0 ). Consequently, the sign of s2 depends on the parameter position with respect to the cusps. More precisely, Z < 0 on the right branch of k and Z > 0 on the left one. It follows that s2 has the sign of b on the left branch of k , and the sign of −b on the right branch. As a conclusion, if the potential V is hard (b < 0) the homoclinic solutions of the truncated normal form described above exist for parameter values near the left branch of each “tongue” k restricted to 0 . If V is soft (b > 0), homoclinic solutions exist for parameter values near the right branch. We sum up the situation in Fig. 2. 5.4.2. General case (a = 0). We now consider the general case a = 0. We introduce the parameter η = ab2 and recall the expression of s2 , (1 −
2T02 q0 ). (93) )s2 = T02 a 2 (3η + 4 − 2 tan(q0 /2) 2γ0 T02 cos(q0 ) − T02 (1 + 2γ0 ) + 4q02
One can obtain a simpler expression for s2 . Indeed, one can prove the identity q02 = T02 (1 + 2γ0 ) − 2 cos (q0 /2) (−1)m T02 γ0
(94)
using successively
γ
− − − Γ1 − − − − − − − − − − − − − − − −−
Γ2
Existence of homoclinic orbits for b > 0
Γ4
− −
.. .. .. .. .. ..
− − − − − − − − − − −
Γ5
− − − − − − − − − −
.. .. .. .. ..
−− − − −−
.. .. .. ... .. .. ..
− − −− − − − − − − − − −
− − − − − −
−
0
...
Γ3
2π
4π
6π
−− −− − −
− − − − − − − − − − − − − −
.. .. .. .. ...
− −
8π
− − − −
.. .. .. .. .. .. .. .. . .. 10π
− −− Existence of homoclinic orbits − for b < 0
T
Fig. 2. Regions in the parameter space where small amplitude homoclinic orbits exist in the case a = 0 (even potentials)
78
G. James, Y. Sire
q02 = −4
q0 + T02 (1 + 2γ0 ) tan(q0 /2)
(95)
(see Eq. (31)) and 2q0 = (−1)m T02 γ0 sin(q0 /2)
(96)
(see Eqs. (28)–(30)). Identity (94) allows us to simplify the right side of (93). Indeed, we obtain by substitution 2γ0 T02 cos(q0 )−T02 (1+2γ0 )+4q02 = T02 (3+6γ0 +2γ0 cos(q0 )−8γ0 (−1)m cos(q0 /2)), which simplifies in 2γ0 T02 cos(q0 ) − T02 (1 + 2γ0 ) + 4q02 = T02 (3 + 16γ0 sin4 (
mπ q − )). 4 2
Consequently, one can write s2 in the form (1 −
2 q0 )s2 = T02 a 2 (3η + 4 − 2 tan(q0 /2) 3 + 16γ0 sin4 ( q40 −
mπ ) 2 )
for (γ0 , T0 ) ∈ m .
We study the sign of s2 when (T0 , γ0 ) covers the left or the right branch of the “tongue” l of such that Z(q ) > 0 (left m . To this end, we fix m ≥ 1 and introduce the subset m m 0 r of such that Z(q ) < 0 (right branch). Note that (T , γ ) ∈ branch) and the subset m m 0 0 0 l is equivalent to q ∈ (q, m ¯ qmax ), where q¯ ∈ (2mπ, 2(m + 1)π ) denotes the point 0 satisfying Z(q) ¯ = 0 (corresponding to the cusp of m ) and qmax ∈ (2mπ, 2(m + 1)π ) is obtained by fixing T = 0 in Eq. (27) or (29) (γ goes to infinity and T = 0 at this value r is equivalent to fixing q ∈ (2mπ, q) of q). Similarly, having (T0 , γ0 ) ∈ m ¯ (see Fig. 3). 0 q
q max
γ
q
2mπ
r Γ m
− γ( q) − ) ( T(q),
l Γm T
l and r Fig. 3. Definition of m m
Travelling Breathers in Klein-Gordon Chains
79
We denote by Fm the quantity Fm (q0 ) = 4 − and s2 writes
1−
2 3 + 16γ0 (q0 ) sin4 ( q40 −
mπ , 2 )
q0 s2 = T02 a 2 (3η + Fm (q0 )). 2 tan(q0 /2)
(97)
(98)
Note that Fm is a strictly increasing function of q0 for q0 ∈ (2mπ, qmax ). Moreover, we 10 have Fm (2mπ ) = 10 ¯ < 4. We deduce the following 3 , Fm (qmax ) = 4 and 3 < Fm (q) results. l (m ≥ 1) • Case (T0 , γ0 ) ∈ m In this case, we have
sign(s2 ) = sign(3η + Fm (q0 )).
(99)
¯ l (this is the case in particular for b ≥ 0). For If η > − Fm3(q) , we have s2 > 0 on m Fm (q) ¯ 4 l . Finally, if η < − 4 then s − 3 < η < − 3 , s2 is negative only on a piece of m 2 3 l . is negative on m r (m ≥ 1) • Case (T0 , γ0 ) ∈ m In this case, we have
sign(s2 ) = −sign(3η + Fm (q0 )).
(100)
r If η > − 10 9 , we have s2 < 0 on m (this is the case in particular for b ≥ 0). For Fm (q) ¯ ¯ 10 r . Finally, if η < − Fm (q) − 3 < η < − 9 , s2 is negative only on a piece of m 3 r then s2 is positive on m .
We illustrate our analysis for the particular curve 1 (the other curves yield qualitatively similar results). Figure 4 describes the sign of s2 depending on q0 and η. Figure 5 indicates the regions on 1 where s2 < 0. We recall that the homoclinic solutions (87) exist for s2 (γ0 , T0 ) < 0, (T0 , γ0 ) ∈ 0 and (T , γ ) ≈ (T0 , γ0 ) outside of the “tongue” k . 6. Homoclinic Solutions for the Klein-Gordon System In this section we construct approximate (leading order) travelling breather solutions of the Klein-Gordon system with reversible homoclinic solutions of the truncated normal form. In addition we obtain exact solutions in the case of even potentials. We choose (γ , T ) ≈ (γ0 , T0 ) ((T0 , γ0 ) ∈ 0 ), in such a way that the linearized operator L has four symmetric eigenvalues close to ±iq0 and having non-zero real parts. In addition we require s2 (γ0 , T0 ) < 0. In this case, the truncated normal form admits different types of homoclinic solutions (A, B, C, D) described in Sect. 5.3. In the sequel we restrict our attention to reversible solutions under R or R1 = R S, for which a persistence theory has been developed [28]. According to (76), reversible approximate solutions of (11) are given by U ≈ AV0 + B Vˆ0 + CV1 + DV2 + c.c.,
(101)
80
G. James, Y. Sire η
− q
2π
q
max q
s −10/9
2
0
< 0 s
2
> 0
−1.15
s
2
> 0 s
−4/3
2
< 0
Fig. 4. Sign of s2 in the case of general potentials for (T0 , γ0 ) ∈ 1 . Note that in this case qmax ≈ 11.2 and q¯ ≈ 9
γ
γ
− − − − −− −− −− −
− − − − − − − − −
Case
η > −10/9
T
− Case −F ( q ) /3 < η < −10/9 1 γ
γ
− −− − − −− −− − −− Case −4/3 < η < −F ( q− ) /3 1
T
T
− − − − − − − − − − − − − − − − −− −− −− Case
η < −4/3
T
Fig. 5. Parts of 1 where s2 < 0 (bold line). The dashed regions correspond to the existence of small ¯ amplitude homoclinic solutions given by (87). Note that − F13(q) ≈ −1.15
where A, B, C, D have the form (87). One fixes θ, θ1 , θ2 equal to 0 or π if U is reversible under R. If (T0 , γ0 ) ∈ 2k+1 and U is reversible under R1 , one has θ, θ1 = ±π/2 and θ2 equal to 0 or π. For (T0 , γ0 ) in m ∩ 0 , (101) yields the approximate solutions of (7),
Travelling Breathers in Klein-Gordon Chains
u1 (t) u2 (t)
≈ A(t)
(−1)m 1
81
+ C(t)
−1 1
+ D(t)
1 + c.c. 1
Coming back to the original variables (using Eq. (6)), we obtain xn (τ ) ≈ [ (−1)nm A + (−1)n C + D ] ( As ξ =
τ T
−
n−1 2
n−1 τ − ) + c.c. T 2
(102)
→ ±∞ one has
A(ξ ) ∼ A0 e−a |ξ | ei(qˆ0 ξ ±φ+θ) , C(ξ ) ∼ β ei(qˆ1 ξ ±φ1 +θ1 ) , D(ξ ) ∼ α ei(qˆ2 ξ ±φ2 +θ2 ) with a > 0. Approximate solutions given by (102) converge for C, D = 0 towards quasiperiodic solutions as ξ → ±∞, and have larger oscillations at the center for α, β |A(0)|. Homoclinic solutions bifurcating in the neighborhood of 2m ∩ 0 can be seen as superpositions of a travelling wave of permanent form xT W (τ ) = (A+D)( Tτ − n−1 2 ) and τ n−1 n a pulsating travelling wave xT P (τ ) = (−1) C( T − 2 ). If β |A(0)|, the pulsating part xT P is mainly visible at the wave tail. Note that pure travelling waves (with C = 0, D = 0) exist in the full system (1) [22]. In addition, homoclinic solutions bifurcating in the neighborhood of 2m+1 ∩ 0 can be seen as superpositions of a pulsating travelling wave xT P (τ ) = (−1)n (A + τ n−1 C)( Tτ − n−1 2 ) and a travelling wave of permanent form xT W (τ ) = D( T − 2 ). For α, β |A(0)|, the wave mainly consists of a O(|A(0)|) pulsating part localized at the center and a small quasiperiodic tail. If we fix γ = γ0 and expand (102) for δ = |T − T0 | ≈ 0, α ≈ 0, β ≈ 0 we obtain for bounded values of τ, n, ˆ 1/2 (n − vg τ )) ei(k0 n−ω0 τ ) + β ei(k1 n−ω1 τ ) + α ei(k2 n−ω2 τ ) + c.c., xn+1 (τ ) ≈ δ 1/2 A(δ (103) where k0 = q20 − mπ , ω0 = Tq00 , k1 = q21 − π , k2 = q22 , ωi = Tq0i , vg = T20 , Aˆ has ˆ ) = c1 (cosh(c2 ξ ))−1 and phase shifts have been included in c1 , α, β for the form A(ξ notational simplicity. We note that (ωi , ki ) satisfies the equation ω2 = 1 + 4γ0 sin2
k 2
(104)
due to the fact that qi satisfies the dispersion relation (19). One recognizes in Eq. (104) the usual form of the dispersion relation of Eq. (1) linearized at xn = 0. Moreover Eq. (103) shows that our approximate solutions can be seen as superpositions of modulated plane waves, and one can check that vg is the group velocity ω (k0 ) (use Eqs. (28), (30)). Note that, due to condition (2) (p = 2), only specific wave vectors k1 , k2 in the oscillatory tail are selected among the whole set of possible ones. Without further symmetry assumptions (evenness of V , or restriction to travelling wave solutions with C = 0 as in [22]), the persistence of solutions (102) for Eq. (1) is still an open problem, which should be tackled using the finite dimensional reduced system (70). From the analysis of Sect. 5.3, we conjecture that the particular reversible solutions decaying to 0 at infinity (C = D = 0) should not persist generically in the Klein-Gordon system (1). To make a more precise statement, fix V (x) = 21 x 2 − a3 x 3 − b4 x 4 and assume
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(T , γ ) close to 0 . We conjecture that a solution of (11) reversible under R or R1 , homoclinic to 0 and close to an approximate solution (101) with C = D = 0 might only exist if (T , γ , a, b) is chosen on a discrete collection of codimension-l submanifolds of R4 (l > 0). The codimension depends on the number of pairs of purely imaginary eigenvalues (i.e. the number of resonant phonons) in our parameter regime and symmetry assumptions. In the present case (with two pairs of purely imaginary eigenvalues, in addition to weakly hyperbolic ones), we expect l = 2 if (T0 , γ0 ) ∈ 2m+1 ∩ 0 (case of travelling breather solutions) and l = 1 when (T0 , γ0 ) ∈ 2m ∩ 0 (case of solitary wave solutions, which have the additional invariance under S). The codimension is equal to the number of compatibility conditions obtained with the normal form (70) for each type of homoclinic bifurcation (see Sect. 5.3). Instead of solutions decaying to 0 at infinity, we conjecture for (T0 , γ0 ) ∈ 2m+1 ∩ 0 the persistence of reversible solutions homoclinic to quasi-periodic waves (since we conjecture the persistence of reversible homoclinic orbits to 2−dimensional tori in the normal form (70)). Reversible approximate solutions (102) should constitute the principal part of travelling breather solutions of (1) superposed on a small quasi-periodic oscillatory tail. The following theorem summarizes the above results in the case of travelling breather solutions. Theorem 6.1. Assume s2 (γ0 , T0 ) < 0 defined by Eq. (77) for a fixed (T0 , γ0 ) ∈ 0 2k+1 and consider (γ , T ) ≈ (γ0 , T0 ) such that the linear operator L in (11) has four symmetric eigenvalues close to ±iq0 and having non-zero real parts. Then the reduced Eq. (69) written in the normal form (70) and truncated at order 4 admits small amplitude reversible solutions (under R or R S) homoclinic to 2-tori. Such solutions should correspond to the principal part of travelling breather solutions of system (1), superposed at infinity on an oscillatory (quasiperiodic) tail, and given at leading order by the expression τ n−1 − ) + c.c, (105) T 2 where A, C, D are defined in Eq. (87) (with θ2 equal to 0 or π , θ, θ1 = ±π/2 for R S-reversible solutions, and θ, θ1 equal to 0 or π for R-reversible solutions). xn (τ ) ≈ [ (−1)n A + (−1)n C + D ] (
In addition to leading order approximate solutions, we obtain exact travelling breather solutions superposed on a small oscillatory tail in the case of even potentials. This result follows directly from the center manifold reduction theorem (Theorem 4.1) and the analysis of the reduced equation (see Sect. 5.3.3). Theorem 6.2. Assume s2 (γ0 , T0 ) < 0 defined by Eq. (77) for a fixed (T0 , γ0 ) ∈ 0 2k+1 and consider (γ , T ) ≈ (γ0 , T0 ) such that the linear operator L in (11) has four symmetric eigenvalues close to ±iq0 and having non-zero real parts. Moreover assume that the potential V is even. Equation (11) is invariant under the symmetry −S defined in (17). If (T0 , γ0 ) lies outside some subset of 0 2k+1 having zero Lebesgue measure (corresponding to resonant cases), the full reduced Eq. (69) restricted to Fix(−S) admits small amplitude reversible solutions (under ±R) homoclinic to periodic orbits. These solutions correspond to exact travelling breather solutions of system (1) superposed at infinity on an oscillatory (periodic) tail. Their principal part is given by xn (τ ) = (−1)n [ A + C ] (
n−1 τ − ) + c.c. + h.o.t, T 2
(106)
Travelling Breathers in Klein-Gordon Chains
83
where A, C are given by Eq. (87) (with θ, θ1 = ±π/2 for reversible solutions under −R, and θ, θ1 equal to 0 or π for reversible solutions under R). For a fixed value of (γ , T ) (and up to a time shift), these solutions occur in a one-parameter family parametrized by the amplitude β of oscillations at infinity. The lower bound of these amplitudes is 1/2 O(e−c/µ ), where µ = |T − T0 | + |γ − γ0 |, c > 0. Remark. The lower bound of the amplitudes should be generically nonzero, but may vanish on a discrete collection of curves in the parameter plane (T , γ ). As a consequence, in a given system (1) (with fixed coupling constant γ and symmetric on-site potential V ), exact travelling breather solutions decaying to 0 at infinity (and satisfying (2) for p = 2) may exist in the small amplitude regime, for isolated values of the breather velocity 2/T . We conclude by comparing our findings to a previous work. The existence of modulated plane waves in Klein-Gordon chains has been studied by Remoissenet [36] using formal multiscale expansions. Under this approximation, the wave envelope satisfies the nonlinear Schr¨odinger (NLS) equation. In this problem a rigorous analysis of the validity of NLS equation (on large but finite time intervals) has been performed in [19]. The condition obtained by Remoissenet for the existence of NLS solitons (for the specific wave number k = k0 = q20 − mπ ) is exactly the condition s2 < 0 derived in Sect. 5.3. Indeed, the condition obtained by Remoissenet is P Q > 0, where Q=
T0 2a 2 (4a 2 − + 3b), 2q0 3 + 16γ0 sin4 ( k20 )
(107)
γ0 T 2 γ0 T0 (cos(k0 ) − 20 sin2 (k0 )). 2q0 q0
(108)
P =
Using the same equations as in Sect. 5.4.2 one can express P and Q as a function of γ0 , T0 , q0 , Q=
2a 2 T02 T0 (4a 2 − + 3b), 2q0 −T02 (1 + 2γ0 ) + 2γ0 T02 cos(q0 ) + 4q02
(109)
1 (−4 + γ0 T02 (−1)m cos(q0 /2)). 2q0 T0
(110)
P =
The coefficient P is Z(q0 ) (defined in (91)) multiplied by a negative constant (use Eq. (96)). Similarly, the expression into brackets in Q is exactly the same as the one in the normal form coefficient s2 . Consequently, the product P Q differs from s2 by a negative multiplicative factor, and thus P Q > 0 is equivalent to s2 < 0. Acknowledgements. We wish to thank G´erard Iooss for helpful comments. We are grateful to Serge Aubry for his hospitality at the Laboratoire L´eon Brillouin (CEA Saclay, France) and stimulating discussions. This work has been supported by the European Union under the RTN project LOCNET (HPRN-CT1999-00163).
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References 1. Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17, 1011–1018 (1976) 2. Ablowitz, M.J., Musslimani, Z., Biondini, G.: Methods for discrete solitons in nonlinear lattices. Phys. Rev. E 65, D56618-1–13 (2002) 3. Ablowitz, M.J., Musslimani, Z.: Discrete spatial solitons in a diffraction-managed nonlinear waveguide array : a unified approach. Physica D 184, 276–303 (2003) 4. Aigner, A.A., Champneys, A.R., Rothos, V.M.: A new barrier to the existence of moving kinks in Frenkel-Kontorova lattices. Physica D 186, 148–170 (2003) 5. Aubry, S., Kopidakis, G., Kadelburg, V.: Variational proof for hard discrete breathers in some classes of hamiltonian dynamical systems. Discrete and Continuous Dynamical Systems B 1, 271–298 (2001) 6. Aubry, S., Cr´et´egny, T.: Mobility and reactivity of discrete breathers. Physica D 119, 34–46 (1998) 7. Berger, A., MacKay, R.S., Rothos, V.M.: A criterion for non-persistence of travelling breathers for perturbations of the Ablowitz-Ladik lattice. Discrete Cont. Dyn. Sys. B. 4, no. 4, 911–920 (2004) 8. Bickham, S.R., Kiselev, S.A., Sievers, A.J.: Stationary and moving intrinsic localized modes in one-dimensional monoatomic lattices with cubic and quartic anharmonicity. Phys. Rev. B 47, 21 (1993) 9. Dauxois, T., Peyrard, M., Willis, C.R.: Discreteness effects on the formation and propagation of breathers in nonlinear Klein-Gordon equations. Phys. Rev. E 48, 4768 (1993) 10. Duncan, D.B., Eilbeck, J.C., Feddersen, H., Wattis, J.A.D.: Solitons in lattices. Physica D 68, 1–11 (1993) 11. Eilbeck, J.C., Flesch, R.: Calculation of families of solitary waves on discrete lattices. Physics Letters A 149, 200–202 (1990) 12. Flach, S., Kladko, K.: Moving discrete breathers ? Physica D 127, 61–72 (1999) 13. Flach, S., Willis, C.R.: Movability of localized excitations in Nonlinear Discrete systems: a separatrix problem. Phys. Rev. Lett. 72, 1777–1781 (1994) 14. Flach, S., Zolotaryuk, Y., Kladko, K.: Moving kinks and pulses: an inverse method. Phys. Rev. E 59, 6105–6115 (1999) 15. Flach, S., Willis, C.R.: Discrete Breathers. Phys. Rep. 295, 181–264 (1998) 16. Friesecke, G., Matthies, K.: Atomic-scale localization of high-energy solitary waves on lattices. Physica D 171, 211–220 (2002) 17. Friesecke, G., Pego, R.L.: Solitary waves on FPU lattices : I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12, 1601–1627 (1999) 18. Friesecke, G., Wattis, J.A.: Existence theorem for solitary waves on lattices. Commun. Math. Phys. 161, 391–418 (1994) 19. Giannoulis, J., Mielke, A.: The nonlinear Schr¨odinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities. Nonlinearity 17, 551–565 (2004) 20. Iooss, G.: Travelling waves in the Fermi-Pasta-Ulam lattice. Nonlinearity 13, 849–866 (2000) 21. Iooss, G., Adelmeyer, M.: Topics in bifurcation theory and applications. Adv. Ser. Nonlinear Dyn 3, Singapore: World Sci. (1998) 22. Iooss, G., Kirchg¨assner, K.: Travelling waves in a chain of coupled nonlinear oscillators. Commun. Math. Phys. 211, 439–464 (2000) 23. Iooss, G., Lombardi, E.: Polynomial normal forms with exponentially small remainder for analytic vector fields, To appear in J. Diff. Eqs. Preprint Institut Non Lineaire de Nice, 2004 24. Iooss, G., P´erou`eme, M-C.: Perturbed homoclinic solutions in reversible 1:1 resonance vertor fields. J. Diff. Eqs. 102, 62–88 (1993) 25. James, G.: Centre manifold reduction for quasilinear discrete systems. J. Nonlinear Sci 131 , 27–63 (2003) 26. Kastner, M., Sepulchre, J-A.: Effective Hamiltonian for traveling discrete breathers in the FPU chain. Submitted to Discrete Cont. Dyn. Sys. B (2003) 27. Kirchg¨assner, K.: Wave solutions of reversible systems and applications. J. Diff. Eqs. 45, 113–127 (1982) 28. Lombardi, E.: Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems. Lecture Notes in Mathematics, Vol. 1741, Berlin-Heidelberg-Newyork: Springer-Verlag, 2000 29. Lombardi, E.: Phenomena beyond all orders and bifurcations of reversible homoclinic connections near higher resonances. In: Peyresq Lectures on Nonlinear Phenomena. Kaiser R., Montaldi, J. (eds.), Singapore: World Scientific, 2000 p. 161–200 30. Mackay, R.S.,Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994)
Travelling Breathers in Klein-Gordon Chains
85
31. MacKay, R.S., Sepulchre, J-A.: Effective Hamiltonian for travelling discrete breathers. J. Phys. A 35, 3985–4002 (2002) 32. Mallet-Paret, J.: The global structure of traveling waves in spatially discrete systems, J. Dyn. Diff. Eqs. 11, 99–127 (1999) 33. Mielke, A.: Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Aci. 10, 51–66 (1988) ¨ 34. Mielke, A.: Uber maximale Lp -Regularit¨at f¨ur Differentialgleichungen in Banach und Hilbert Ra¨umen. Math. Ann. 277, 121–133 (1987) 35. Morgante, A.M., Johansson, M., Kopidakis, G., Aubry, S.: Standing wave instabilities in a chain of nonlinear coupled oscillators. Physica D 162, 53–94 (2002) 36. Remoissenet, M.: Low-amplitude breather and envelope solitons in quasi-one-dimensional physical models. Phys. Rev. B 33, number 4, 2386–2392 (1986) 37. Sandusky, K.W., Page, J.B., Schmidt, K.E.: Stability and motion of intrinsic localized modes in nonlinear periodic lattices. Phys. Rev. B 46, 10, 6161–6168 (1992) 38. Savin, A.V., Zolotaryuk, Y., Eilbeck, J.C.: Moving kinks and nanopterons in the nonlinear KleinGordon lattice. Physica D 138, 267–281 (2000) 39. Sepulchre, J-A.: Energy barriers in coupled oscillators: from discrete kinks to discrete breathers. In: Proceedings of the Conference on Localization and Energy Transfer in Nonlinear Systems, June 17-21, 2002, San Lorenzo de El Escorial, Madrid, Spain; eds. L. Vazquez, R.S. MacKay, M-P. Zorzano, Singapore: World Scientific. (2003), pp. 102–129 40. Sire, Y., James, G.: Travelling breathers in Klein-Gordon chains. C. R. Acad. Sci. Paris, Ser. I 338, 661–666 (2004) 41. Smets, D., Willem, M.: Solitary waves with prescribed speed on infinite lattices. J. Funct. Anal. 149, 266–275 (1997) 42. Szeftel, J., Huang, G., Konotop, V.: On the existence of moving breathers in one-dimensional anharmonic lattices. Physica D 181, 215–221 (2003) 43. Takeno, S., Hori, K.: A propagating self-localized mode in a one-dimensional lattice with quartic anharmonicity. J. Phys. Soc. Japan 59, 3037–3040 (1990) 44. Sievers, A.J., Takeno, S.: Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970–973 (1988) 45. Tsurui, A.: Wave modulations in anharmonic lattices. Progr. Theor. Phys 48, no 4, 1196–1203 (1972) 46. Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimensions. Dynamics Reported 1, new series, 125–163 (1992) Communicated by A. Kupiainen
Commun. Math. Phys. 257, 87–117 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1262-9
Communications in
Mathematical Physics
Dispersive Estimates for Schr¨odinger Operators in Dimension Two W. Schlag Division of Astronomy, Mathematics, and Physics, 253-37 Caltech, Pasadena, CA 91125, USA. E-mail:
[email protected] Received: 21 April 2004 / Accepted: 16 July 2004 Published online: 11 January 2005 – © Springer-Verlag 2005
Abstract: We prove L1 (R2 ) → L∞ (R2 ) for the two-dimensional Schr¨odinger operator −+V with the decay rate t −1 . We assume that zero energy is neither an eigenvalue nor a resonance. This condition is formulated as in the recent paper by Jensen and Nenciu on threshold expansions for the two-dimensional resolvent. 1. Introduction The purpose of this paper is to prove the following result. Theorem 1. Let V : R2 → R be a measurable function such that |V (x)| ≤ C(1 + |x|)−β , β > 3. Assume in addition that zero is a regular point of the spectrum of H = − + V . Then itH e Pac (H )f ≤ C|t|−1 f 1 ∞ for all f ∈ L1 (R2 ). The definition of zero being a regular point amounts to the following, see Jensen, 1 Nenciu [JenNen] and Definition 7 below: Let V ≡ 0 and set U = sign V , v = |V | 2 . Let Pv be the orthogonal projection onto v and set Q = I − Pv . Finally, let 1 log |x − y| f (y) dy. (G0 f )(x) := − 2π R2 Then zero is regular iff Q(U + vG0 v)Q is invertible on QL2 (R2 ). Jensen and Nenciu study ker[Q(U + vG0 v)Q] on QL2 (R2 ). It can be completely described in terms of solutions of H = 0. In particular, its dimension is at most
The author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship
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W. Schlag
three plus the dimension of the zero energy eigenspace, see Theorem 6.2 and Lemma 6.4 in [JenNen]. The extra three dimensions here are called resonances. Hence, the requirement that zero is a regular point is the analogue of the usual condition that zero is neither an eigenvalue nor a resonance of H . As far as the spectral properties of H are concerned, we note that under the hypotheses of Theorem 1 the spectrum of H on [0, ∞) is purely absolutely continuous, and that the spectrum is pure point on (−∞, 0) with at most finitely many eigenvalues of finite multiplicities. The latter follows for example from Stoiciu [Sto], who obtained Birman-Schwinger type bounds in the case of two dimensions. Theorem 1 appears to be the first L1 → L∞ bound with |t|−1 decay in R2 . Yajima [Yaj] and Jensen, Yajima [JenYaj] proved the Lp (R2 ) boundedness of the wave operators under stronger decay assumptions on V (x), but only for 1 < p < ∞. Hence their result does not imply Theorem 1. Local L2 decay was studied by Murata [Mur], but he does not consider L1 → L∞ estimates. n The first L1 (Rn ) → L∞ (Rn ) bounds for eitH with |t|− 2 decay were obtained by Journ´e, Soffer, and Sogge [JouSofSog]. However, their argument depends on the fact n that t − 2 is integrable at t = ∞, and thus only applies for n ≥ 3. In dimension n = 1 1 Weder [Wed] obtained the |t|− 2 -decay under some conditions on V which were then relaxed by Goldberg and the author [GolSch]. However, the case n = 2 remained open. As usual, the proof of Theorem 1 breaks up into two regimes: energies bigger than λ1 and energies in (0, λ1 ). Here λ1 > 0 is some small constant. The corresponding statements are Propositions 4 and 11 below. Theorem 1 then follows by combining these two propositions. For energies in (0, λ1 ) we use the recent results of Jensen and Nenciu [JenNen] on expansions of the resolvent (H − (λ2 ± i0))−1 for λ close to zero. Since we require somewhat finer estimates on various error terms, we give a complete derivation of this expansion. However, we emphasize that this derivation is of course merely a variant of a special case of the expansions in [JenNen]. In fact, the main achievement of Jensen and Nenciu is to determine the expansion of the perturbed resolvent in the presence of a resonance and/or an eigenvalue at zero. 2. Energies Separated from Zero The main purpose of this section is to prove the dispersive estimate for the evolution restricted to energies [λ1 , ∞), λ1 > 0. This will be accomplished by an expansion of the perturbed resolvent into a finite Born series, see (18) and (19). The main difficulty is to obtain the dispersive bound for each term of the Born series. This is done in Lemma 3 below. For the remainder (19) in the Born expansion, which still contains the perturbed resolvent, we use the limiting absorption principle. The approach in this section is modelled after that in [GolSch], which in turn had its origins in the work of Rodnianski and the author [RodSch]. Lemma 2 is a variant of the standard stationary phase method. In what follows, the notation x y means that x ≤ Cy for some constant C. Lemma 2. Let φ(0) = φ (0) = 0 and 1 ≤ φ
≤ C. Then ∞ |a (x)| |a(x)| itφ(x) δ2 e a(x) dx + χ[|x|>δ] dx , 2 2 δ + |x| |x| −∞ 1
where δ = |t|− 2 .
(1)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
89
Proof. With η being a standard cut-off one has ∞ ∞ eitφ(x) a(x) dx ≤ eitφ(x) a(x)η(x/δ) dx −∞ −∞ ∞ + eitφ(x) a(x)(1 − η(x/δ)) dx −∞ a(x)(1 − η(x/δ)) 2 |a(x)| dx + δ dx φ (x) |x|<δ |a(x)| |a (x)| |a(x)| dx + δ 2 + dx, |x|2 |x| |x|<δ |x|δ as claimed.
It is well-known that i R0± (λ2 )(x, y) = (− − (λ2 ± i0))−1 (x, y) = ± H0± (λ|x − y|), 4 where H0± are the Hankel functions of order zero with H0− = H0+ . They have the form H0+ (y) = ei(y−1) ω(y)χ[y>1] + ω(y)χ[0
and satisfy the bounds |ω(y)| |y|− 2 if y 1 and |ω(y)| | log y| of 0 < y < 21 . Moreover, one has for all positive integers ν, 1
|ω(ν) (y)| |y|−ν− 2 if y 1 , |ω(ν) (y)| |y|−ν if 0 < y < 1. Set ω+ (y) = χ1 (y/y0 )ω(y), where χ1 (y) = 0 if y ≤ 1 and = 1 if y ≥ 2. Here y0 1 is a fixed constant. Define ω− (y) via ω = ω+ + ω− , i.e., ω− (y) = (1 − χ1 (y/y0 ))ω(y) (in Sect. 3 the functions ω+ and ω− will take on a different meaning, not to be confused with the one here). Let 2 V K := sup (2) 1 + log− |x − y| |V (y)| dy, x∈R2 R2
where log− u = −χ[0
∈J ∗
0
j ∈J
m−1 ω− (λ|x −1 − x |) dλ |V (xk )| dx1 . . . dxm−1 |t|−1 V m−1 K k=1
with a constant that only depends on m.
(3)
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W. Schlag
Proof.The heuristic reason for this bound is as follows: Let dj = |xj − xj −1 | and s = j ∈J dj . If there is a critical point of the phase, then it is λ0 = 2ts (assuming t > 0). We may assume that λ0 1, otherwise the integrand vanishes at λ0 . Using stationary phase, the inner integral is then bounded by
1
1
t − 2 λ0 (λ0 s)− 2
log− (λ0 d ) t −1
∈J ∗
log− (d ).
∈J ∗
Inserting this bound into (3) then yields the desired result by an application of CauchySchwartz, see (5) below. To make this rigorous, we start off integrating by parts: Then |t|
∞
2 ±λs)
λ ei(tλ
χ1 (λ)χ2 (λ/L)
0
ω+ (λdj )
∈J ∗
j ∈J ∞
2 ±λs)
ei(tλ
0
χ1 (λ)χ2 (λ/L)
ω− (λd ) dλ
ω+ (λdj )
ω− (λd ) dλ
∈J ∗
j ∈J
1 ∞ i(tλ2 ±λs) e χ1 (λ)χ2 (λ/L) ω+ (λdj ) ω− (λd ) dλ + L 0 j ∈J
∈J ∗ ∞ 2 ei(tλ ±λs) χ1 (λ)χ2 (λ/L) ω+ (λdj ) ω− (λd ) dλ +s 0
+
dk
k∈J
+
j ∈J ∞
e
i(tλ2 ±λs)
0
dk
k∈J ∗
∈J ∗
∞
χ1 (λ)χ2 (λ/L) ω+ (λdk )
2 ±λs)
ei(tλ
χ1 (λ)χ2 (λ/L)
0
j ∈J
ω+ (λdj )
(4) ω− (λd ) dλ
∈J ∗
j ∈J j =k
ω+ (λdj ) ω− (λdk )
ω− (λd ) dλ
∈J ∗
=k
=: A± + B ± + C ± + D ± + E ± . Let k(x, y) := 1 + log− |x − y|. Then since log− is decreasing,
|A± |
|χ1 (λ)|
|ω− (λdj )| dλ
j ∈J ∗
|χ1 (λ)|
j ∈J ∗
(1 + log− (λdj )) dλ
j ∈J ∗
k(xj −1 , xj ).
|χ1 (λ)|
j ∈J ∗
(1 + log− (dj )) dλ
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
91
Hence the contribution of A± to (3) is
m
R2(m−1) j =1
k(xj −1 , xj )
m−1
|V (xj )| dx1 . . . dxm−1
j =1
m−1 k 2 (x0 , x1 )|V (x1 )| + |V (x1 )|k 2 (x1 , x2 ) |V (xj )|k(xj , xj +1 ) dx1 . . . dxm−1 j =2
V K . m−1
(5)
For the remainder of the proof we set
P∗ =
k(xj −1 , xj )
j ∈J ∗
with the understanding that P∗ = 1 if J ∗ = ∅. Similarly, for L ≥ 1, one has that 1 ∞
± |χ2 (λ/L)| |ω− (λdj )| dλ P∗ . |B | L 0 ∗ j ∈J
± ± Hence the contribution by B ± to (3) is again V m−1 K . The terms D , E are also easy to deal with. Indeed, one has ∞ 3 1 dk (1 + λdk )− 2 (1 + λdj )− 2 dλ P∗ |D ± | 0
k∈J
∞
=
−2
0
j ∈J j =k 1
(1 + λdj )− 2
dλ P∗ = 2 P∗ .
j ∈J
As far as E ± is concerned, we conclude similarly that (with some small constant c > 0) ∞ dk (dk λ)−1 χ[cλdk ≤1] log− (cλdj ) dλ |E ± | 1
k∈J ∗
∞
=
−
1
j ∈J ∗ j =k
log− (cλdj ) dλ P∗ .
j ∈J ∗
We now apply Lemma 2 to C − with φ− (λ) = λ2 − λ st and a(λ) = a+ (λ) ω− (λdj ), a+ (λ) = χ1 (λ)χ2 (λ/L) ω+ (λdj ). j ∈J ∗
Note that |a(λ)| a+ (λ)
(6)
j ∈J
k(xj −1 , xj ) = a+ (λ)P∗ ,
j ∈J ∗
|a (λ)| |a+ (λ)|P∗ +
j ∈J ∗
|a+ (λ)|λ−1 χ[λdj 1]
(7) k∈J ∗ k=j
(1 + log− (λdk )).
(8)
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Set g(y) = 1 + log− (y) so that g (y) = −χ[0
0. Then 0 < b(λ)
j ∈J ∗
(9)
j ∈J ∗
k(xj −1 , xj ) for λ 1 and
(λ)|P∗ + |a+ (λ)||b (λ)| |a+ (λ)|P∗ + a0 (λ)(−b (λ)) , (10) |a (λ)| |a+ 1 where a0 (λ) = χ1 (λ)χ2 (λ/L) j ∈J (1 + λdj )− 2 .
(λ ) = 0 for λ = s . We first assume that χ (λ ) = 0 as well as One has φ− 0 0 1 0 2t λ0 ∈ supp(ω+ (dj ·)) for each j ∈ J . These assumptions translate into λ0 1 and −1 and thus λ 0 minj ∈J λ0 dj 1. The latter condition implies that λ20 = sλ 0 2t t 1
δ = |t|− 2 . Next, we use Lemma 2 to bound |C − |. On the one hand, |a(λ)|P∗−1 2 sδ dλ δ 2 + |λ − λ0 |2 1 1 ∞ λ0 −δ (1 + λs)− 2 (1 + λs)− 2 2 sδ 2 dλ + sδ dλ 2 2 (λ − λ0 )2 λ0 −δ δ + (λ − λ0 ) 1 ∞ √ 2 λ0 −δ 1 dλ 2 − 21 dλ + sδ sδ (1 + λ0 s) √ 2 + (λ − λ )2 δ λ(λ0 − λ)2 0 λ0 −δ 1 1 3 √ − √ − sλ0 2 δ + sδ 2 λ0 2 s − 21 s − 21 −1 (11) λ0 + λ δλ0 1. t t 0 On the other hand, see (10), an integration by parts yields |a (λ)| sδ 2 dλ |λ−λ0 |>δ |λ − λ0 |
(λ)|P |a+ a0 (λ)(−b (λ)) ∗ 2 2 sδ dλ + sδ dλ |λ − λ0 | |λ−λ0 |>δ |λ − λ0 | |λ−λ0 |>δ
(λ)|P |a+ a0 (λ)b(λ) ∗ sδ 2 dλ + sδ 2 dλ |λ−λ0 |>δ |λ − λ0 | |λ−λ0 |>δ |λ − λ0 | λ0 −δ a0 (λ)b(λ) 2 a0 (λ)b(λ) +sδ 2 dλ + sδ . (λ − λ0 )2 |λ − λ0 | λ−λ0 =±δ 1 By the estimates leading up to (11) one has (13) P∗ . On the other hand, ∞ dλ 1 1 1
|χ2 (λ/L)|(1 + λs)− 2 + λ−1 (1 + λs)− 2 (12) sδ 2 P∗ λ − λ0 λ0 +δ L λ0 −δ
|χ2 (λ/L)| 1 +sδ 2 P∗ |χ1 (λ)|(1 + s)− 2 + 1 1 L(1 + λs) 2
3 1 dλ + dj (1 + λdj )− 2 (1 + λdk )− 2 . λ0 − λ j ∈J
k∈J k=j
(12) (13)
(14)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
93
It will be convenient to resum the expression on the right-hand side of (14) by rewriting it as a derivative. This yields (14)
√
−1 sδλ0 2 P∗
+ sδ P∗
P∗ + sδ 2 P∗
+sδ 2 P∗
λ0 −δ
1
P∗ +
sδ P∗
1
1
(1 + λdk )− 2 1
−
1
(1 + λdk )− 2
k∈J
k∈J
2
1
(1 + λdk )− 2
k∈J
√
λ0 −δ
2
dλ λ0 − λ
dλ (λ0 − λ)2
1 λ0 − λ λ=1
λ0 −δ
dλ 1 2
λ (λ0 − λ)2
+ sδ 2 λ−1 0 P∗
√ √ −1 −3 P∗ + ( sδ 2 λ0 2 + sδλ0 2 )P∗ P∗ . In view of the preceding, |C − | P∗ provided λ0 1 and minj ∈J λ0 dj 1. This gives the desired contribution to (3). Now suppose that λ0 1 but minj ∈J λ0 dj 1. Let µ = minj ∈J dj so that µ λ−1 0 . By construction, supp(a) ⊂ [Cµ−1 , ∞) for some large C. Therefore, λ λ0 as well as λ − λ0 λ on supp(a). By Lemma 2,
|a(λ)| |a (λ)|
dλ + 2 [λ0 −δ,λ0 +δ] λ − λ0 λ0 +δ (λ − λ0 )
− 1 √ −1 s δλ0 2 χ[δλ0 ] P∗ + sδ 1 + sµ−1 2 χ[δ λ0 ] P∗ ∞ |a(λ)| |a (λ)|
+sδ 2 + dλ λ2 λ µ−1 ∞ |a(λ)| |a (λ)|
+ dλ. P∗ + sδ 2 λ2 λ µ−1
|C − | sδ
max
|a(λ)| + sδ 2
∞
To bound the integral we use 1
|a(λ)| (1 + sλ)− 2 χ[λ>µ−1 ] P∗ , 1
|a (λ)| λ−1 (1 + sλ)− 2 χ[λ>µ−1 ] P∗ , see (7) and (8). Therefore, sδ 2
µ−1
|a(λ)| λ2
+
1 ∞ (1 + λs)− 2 |a (λ)|
dλ dλ sδ 2 P∗ λ λ2 µ−1 √ 3 sδ 2 µ 2 P∗ sδ 2 µP∗ = λ0 µP∗ P∗ ,
(15)
where we used µ ≤ s to pass to the second inequality in the second line. It remains to consider the case when λ0 1. Note that a(λ) = 0 if minj ∈J λdj 1, which is the
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W. Schlag
same as λ µ−1 . Also, a(λ) = 0 is λ ≤ 1. Then, again via Lemma 2, one obtains as in (15), ∞ √ √ 5 3 |C − | sδ 2 P∗ λ− 2 dλ sδ 2 (1 + µ−1 )− 2 P∗ 1+µ−1
√ √ 3 sδ 2 χ[µ>1] P∗ + sδ 2 µ 2 χ[µ<1] P∗ s s P∗ + µχ[µ<1] P∗ (λ0 + λ0 µχ[µ<1] )P∗ P∗ . t t The lemma is proved.
Proposition 4. Assume that |V (x)| (1 + |x|)−β for some β > 2. Let H = − + V and λ1 > 0 be fixed. Then √ √ sup eitH χ2 ( H /L)χ1 ( H /λ1 ) f, g |t|−1 f 1 g1 L≥1
for all f, g ∈ S(R2 ). The constant here depends only on V and λ1 . Proof. Let RV± (λ2 ) = (− + V − (λ2 ± i0))−1 be the perturbed resolvent. It satisfies the limiting absorption principle, see Agmon [Agm], RV± (λ2 )L2,σ (R2 )→L2,−σ (R2 ) < ∞ ,
(16)
provided σ > 21 . Here L2,σ (R2 ) is the usual weighted space with norm 1 2 f 2,σ = (1 + |x|)2σ |f (x)|2 dx . R2
In addition, one has ∂λ RV± (λ2 )L2,σ (R2 )→L2,−σ (R2 ) < ∞ , provided σ > 23 . The free resolvent satisfies the same bounds with some decay in λ, say λ−α . The exact value of α > 0 is not relevant for our purposes. One has √ √ eitH χ2 ( H /L)χ1 ( H /λ1 ) f, g ∞ dλ 2 . (17) eitλ λ χ2 (λ/L)χ1 (λ/λ1 ) [RV+ (λ2 ) − RV− (λ2 )]f, g = πi 0 We use the resolvent expansion RV± (λ2 ) =
2m+2
R0± (λ2 )(−V R0± (λ2 ))
=0 + R0± (λ2 )(V R0± (λ2 ))m V RV± (λ2 )V (R0± (λ2 )V )m R0± (λ2 ).
Here m is a positive integer that depends on α. Recall that i R0± (λ2 )(x, y) = ± H0± (λ|x − y|) 4
(18) (19)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
95
(the Hankel functions of order zero). By Lemma 3 each of the finitely many terms in (18) leads to the desired time-decay in (17). In fact, this only requires that V K < ∞. For the term (19) one proceeds as in the three-dimensional argument via the limiting absorption principle and stationary phase, see [GolSch]. Following Yajima [Yaj], set G±,x (λ)(x1 ) := e∓iλ|x| R0± (λ2 )(x1 , x). Removing f, g from (17), we are led to proving that ∞ 2 eitλ e±iλ(|x|+|y|) χ2 (λ/L) χ1 (λ/λ1 )λ V RV± (λ2 )V (R0± (λ2 )V )m G±,y (λ), 0 (R0∓ (λ2 )V )m G∗±,x (λ) dλ |t|−1 (20) uniformly in x, y ∈ R2 and L ≥ 1. Next, we check that the derivatives of G+,x (λ) satisfy the estimates (for λ > λ1 > 0) 1 1 j sup ∂λ G±,x (λ) 2,−σ < Cj,σ λ− 2 x−ε provided σ > + j , (21) L 2 x∈R3 1 j (22) sup ∂λ G±,x (λ) 2,−σ < Cj,σ (λx)− 2 provided σ > 1 + j , L
x∈R3
for all j ≥ 0. The small ε > 0 in (21) depends on σ . The bound (22) is Lemma 3.1 in [Yaj]. Alternatively, both bounds follow easily by writing H0± (u) = e±iu ρ± (u), where 1
|ρ± (u)| | log− (u)|χ[0 1 ] . Thus, consider 2
2
2 j ±iλ(|y−x|−|x|) ρ± (λ|x − y|)y−σ 2 2 ∂λ e Ly (R ) y2(j −σ ) |ρ± (λ|x − y|)|2 dy y2(j −σ ) | log(λ|x − y|)|2 dy + λ−1 y2(j −σ ) |y − x|−1 dy R2
[λ|x−y|< 21 ]
2(j −σ ) −2
x
λ
+λ
−1
x
−1
χ[σ >j +1] + λ
−1
x
2(j −σ )+1
χ[σ <j +1] .
The stated bounds now follow by making the appropriate choices of σ depending on j . Rewrite the integral in (20) in the form (with L = ∞) ∞ 2 ± I ± (t, x, y) := eitλ ±iλ(|x|+|y|) ax,y (λ) dλ. (23) 0
By the aforementioned bounds on R0± (λ2 ) and R0± (λ2 ) on weighted L2 -spaces, which ± (λ) has one derivative provide decay in λ, as well as (21), (22), one concludes that ax,y in λ and 1 ± (24) ax,y (λ) (1 + λ)−2 (xy)− 2 for all λ > λ1 , ± for all λ > λ1 , (25) ∂λ ax,y (λ) (1 + λ)−2 x−ε which in particular justifies taking L = ∞ in (23). This requires that one takes m sufficiently large and that |V (x)| (1+|x|)−β for some β > 2. The latter condition arises as
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W. Schlag
follows: Consider, (24). Then by (21) and the limiting absorption principle, respectively, we need to write V = V1 V2 , where V1 decays like x−1−ε , whereas the other should 1 decay like x− 2 −ε . Thus, in this case β > 23 is enough. On the other hand, in (25) one derivative may fall on one of the G-terms at the ends. Then V has to compensate for a 23 + ε power because of (21), and also a 21 + ε power from the limiting absorption principle. Similarly with the other terms. ± (λ) the phase As far as I + (t, x, y) is concerned, note that on the support of ax,y 2 tλ + λ(|x| + |y|) has no critical point. A single integration by parts yields the bound |I + (t, x, y)| t −1 uniformly in x, y ∈ R2 , see (24). In the case of I − (t, x, y) the phase tλ2 − λ(|x| + |y|) has a unique critical point at λ0 = (|x| + |y|)/(2t). If λ0 λ1 , then a single integration by parts again yields the bound of t −1 . If λ0 λ1 then the bound max(|x|, |y|) t is also true, and station1 1 ary phase contributes t − 2 (xy)− 2 t −1 , as desired. To make this rigorous, apply Lemma 2: |I − (t, x, y)| − (λ)| − (λ)| |∂λ ax,y |ax,y −1 |t|−1 dλ + |t| dλ δ 2 + |λ − λ0 |2 [|λ−λ0 |>δ] |λ − λ0 | 1 ∞ (1 + λ)−2 (xy)− 2 (1 + λ)−2 (xy)−ε −1 |t|−1 dλ + |t| dλ δ 2 + |λ − λ0 |2 |λ − λ0 | [|λ−λ0 |>δ] 0 |t|−1 since x+y t. Note that when 0 < t < 1 one has the better bound |I ± (t, x, y)| 1 by (24).
3. Energies Close to Zero The following lemma is a variant of the standard asymptotic expansion around zero energy of the free resolvent on R2 . The estimates on the error terms are written in a somewhat unusual form, which is the one needed later in the proof. Lemma 5. Let R0± (λ2 ) = (− − (λ2 ± i0))−1 be the free resolvent in R2 . Then, for all λ > 0,
i 1 1 γ− log(λ/2) P0 + G0 + E0± (λ). R0± (λ2 ) = ± − 4 2π 2π
(26)
1 Here P0 f := R2 f (x) dx, G0 f (x) = − 2π R2 log |x − y| f (y) dy, and the error E0± (λ) satisfies sup λ− 21 |E ± (λ)(·, ·)| + sup λ 21 |∂λ E ± (λ)(·, ·)| 1 0<λ
0
0<λ
0
with respect to the Hilbert-Schmidt norm in B(L2,s (R2 ), L2,−s (R2 )) with s > 23 .
(27)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
97
Proof. One has, with λ > 0, i R0± (λ2 )(x, y) = ± H0± (λ|x − y|), 4
(28)
where the Hankel functions H0± are H0± (z) = J0 (z) ± iY0 (z) 2 2 = 1 ± i γ ± i log(z/2) + O(z2 log z), π π d 2 ± H0 (z) = ±i + O(z log z). dz πz This is an expansion around z = 0. Around z = ∞ the expansion is given by 2 ± (a(z) ± ib(z))e±i(z−π/4) , H0 (z) = πz with a(z) = 1 −
α z2
± . . . and b(z) =
β z
± . . . . Now let
i
1 1 1 γ− log(λ/2) + log |x − y|. E0± (λ)(x, y) := R0± (λ2 )(x, y) − ± − 4 2π 2π 2π Then |E0± (λ)(x, y)| λ2 |x − y|2 | log(λ|x − y|)|χ[λ|x−y|≤1] + [1 + log(λ|x − y|)]χ[λ|x−y|>1] λ2ε |x − y|2ε | log(λ|x − y|)|χ[λ|x−y|≤1] + [1 + log(λ|x − y|)]χ[λ|x−y|>1] . Hence sup λ−ε |E0± (λ)(x, y)| |x − y|ε .
0<λ
Since the right-hand side has finite Hilbert-Schmidt norm as an operator L2,s (R2 ) → L2,−s (R2 )) with s > 1 + ε, we obtain the first part of (27). On the other hand, λ1−ε |∂λ E0± (λ)(x, y)| 1
1
λ2−ε |x − y|2 | log(λ|x − y|)|χ[λ|x−y|≤1] + [λ 2 −ε |x − y| 2 + λ−ε ]χ[λ|x−y|>1] 1
1
|x − y|ε + λ 2 −ε |x − y| 2 χ[λ|x−y|>1] , and therefore, setting ε = 21 , sup λ 2 |∂λ E0± (λ)(x, y)| |x − y| 2 . 1
1
0<λ
Since the right-hand side has finite Hilbert-Schmidt norm as an operator L2,s (R2 ) → L2,−s (R2 ) with s > 23 , the lemma follows.
98
W. Schlag
Now let V : R2 → R, V ≡ 0, satisfy |V (x)| (1 + |x|)−2β for β > 23 (this condition arises because of the condition s > 23 in the previous lemma). Following Jensen and Nenciu [JenNen] we set U (x) = 1 if V (x) ≥ 0 and U (x) = −1 if V (x) < 0. 1 Also, v(x) := |V (x)| 2 decays like (1 + |x|)−β . The following corollary is therefore an immediate consequence of Lemma 5. Corollary 6. For λ > 0 define M ± (λ) := U + vR0± (λ2 )v. Let P = orthogonal projection onto v. Then
v·,v V 1
M ± (λ) = g ± (λ)P + U + vG0 v + vE0± (λ)v. 1 γ− Here G0 , E0± (λ) are as in Lemma 5 and g ± (λ) = V 1 ± 4i − 2π The remainders satisfy
1 2π
denote the (29) log(λ/2) .
v sup λ− 2 |E0± (λ)(·, ·)| vH S + v sup λ 2 |∂λ E0± (λ)(·, ·)| vH S 1 1
0<λ
1
(30)
0<λ
with respect to the Hilbert-Schmidt norm on L2 (R2 ). The following definition is motivated by [JenNen], cf. the case of S1 = 0 in their Theorem 6.2. Definition 7. Let Q = 1 − P . We say that zero is a regular point of the spectrum of H = − + V provided Q(U + vG0 v)Q is invertible on QL2 (R2 ). In that case set D0 := [Q(U + vG0 v)Q]−1 as an operator on QL2 (R2 ). Jensen and Nenciu show that Q(U + vG0 v) = 0 with ∈ QL2 (R2 ) implies that = U v, where H = 0 in the sense of distributions and ∈ L∞ (R2 ). Thus Definition 7 captures what is sometimes described as absence of zero-energy eigenfunctions and resonances. The following lemma is a technical statement that will be used repeatedly in our argument. Lemma 8. Let D0 be as in Definition 7. Let K be the kernel of the operator QD0 Q. Then the operator with kernel |K| is again L2 -bounded. Proof. For the purposes of this proof we introduce the following terminology:A bounded operator T on L2 (R2 ) is called absolutely bounded if the absolute value of its kernel gives rise to a bounded operator on L2 (R2 ). Note that Hilbert-Schmidt operators are absolutely bounded. Suppose f ∈ QL2 (R2 ) such that QUf = 0, f = 0. Then Uf = cv for some scalar c = 0. Hence f = cU v and Pf = 0 requires that f, v = cU v, v = c R2 V (x) dx = 0. Since this argument can be reversed, ker QL2 (QU Q) = {0} iff V (x) dx = 0.
R2
Case 1. R2 V (x) dx = 0. In this case we claim that QU Q : QL2 (R2 ) → QL2 (R2 ) is invertible. More precisely, one checks that for any g ∈ L2 with Qg = g, f = Ug + c0 U v with c0 = −
Ug, v R2 V (x) dx
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
99
solves QU Qf = g, Qf = f . It is evident from this explicit formula that Q(QU Q)−1 Q is absolutely bounded. Moreover, on QL2 , [Q(U + vG0 v)Q]−1 = (QU Q)−1 [Q + QvG0 v(QU Q)−1 Q]−1 .
(31)
Now vG0 v is a Hilbert-Schmidt operator since v decays faster than (1+|x|)−1−ε . Hence W := QvG0 v(QU Q)−1 Q is also Hilbert-Schmidt. Finally, as an identity on QL2 , [Q + QvG0 v(QU Q)−1 Q]−1 − Q = −[Q + W ]−1 W. Since the right-hand side is Hilbert-Schmidt, we see from (31) that [Q(U + vG0 v)Q]−1 is the composition of an absolutely bounded operator with the sum of an absolutely bounded operator and a Hilbert-Schmidt operator. Hence it is itself absolutely bounded, as claimed. Case 2. R2 V (x) dx = 0. In this case we remark that 0 is an isolated point of the spectrum of QU Q. Let π0 denote the Riesz projection onto ker(QU Q) in QL2 . From the 2 2 preceding, π0 (f ) = V −1 1 f, U vU v. Then QU Q + π0 is invertible on QL (R ). In fact, one checks that an explicit solution of (QU Q + π0 )f = g where Qg = g, Qf = f is given by f = Ug + c1 v − c1 U v with c1 = −
g, U v . R2 |V (x)| dx
In view of this explicit expression, (QU Q + π0 )−1 is absolutely bounded on QL2 . Finally, the identity [Q(U + vG0 v)Q]−1 = [QU Q + π0 + QvG0 vQ − π0 ]−1 on QL2 allows one to repeat the same argument as in Case 1 and the lemma follows. The main technical result in Jensen and Nenciu [JenNen] is a formula for the inverse of M ± (λ)−1 . In the general case, this is complicated, see their Theorem 6.2. But since we are imposing the condition of Definition 7, it is relatively simple to compute that inverse, see the following lemma. Since we need somewhat stronger bounds on the error than those obtained in [JenNen], we give all details. In particular, the proof requires Lemma 8. Lemma 9. Suppose that zero is a regular point of the spectrum of H = −+V . Then for some sufficiently small λ1 > 0, the operators M ± (λ) are invertible for all 0 < λ < λ1 as bounded operators on L2 (R2 ), and one has the expansion M ± (λ)−1 = h± (λ)−1 S + QD0 Q + E ± (λ),
(32)
where h+ (λ) = a log λ + z, a is real, z complex, a = 0, z = 0, and h− (λ) = h+ (λ). Moreover, S is of finite rank and has a real-valued kernel, and E ± (λ) is a HilbertSchmidt operator that satisfies the bound sup λ− 21 |E ± (λ)(·, ·)| + sup λ 21 |∂λ E ± (λ)(·, ·)| 1, (33) HS HS 0<λ<λ1
0<λ<λ1
100
W. Schlag
where the norm refers to the Hilbert-Schmidt norm on L2 (R2 ). Finally, let RV± (λ2 ) = (− + V − (λ2 ± i0))−1 . Then RV± (λ2 ) = R0± (λ2 ) − R0± (λ2 )vM ± (λ)−1 vR0± (λ2 ). This is to be understood as an identity between operators L for some sufficiently small ε > 0.
2, 21 +ε
(34) 2,− 21 −ε
(R2 ) → L
(R2 )
Proof. For the purposes of this proof set T = U + vG0 v. By assumption, QT Q is invertible on QL2 (R2 ). Moreover, by Corollary 6, with respect to the decomposition L2 (R2 ) = P L2 (R2 ) ⊕ QL2 (R2 ), ± g (λ)P + P T P P T Q + vE0± (λ)v. M ± (λ) = QT P QT Q a11 a12 Denote the matrix on the right-hand side by A(λ) = . To invert M ± (λ) and a21 a22 thus A(λ), we use the well-known Fehsbach formula, see e.g. Lemma 2.3 in [JenNen]. −1 This requires that a := (a11 − a12 a22 a21 )−1 exists, and in that case −1 a −aa12 a22 −1 A(λ) = (35) −1 −1 −1 −1 . −a22 a21 a a22 a21 aa12 a22 + a22 Note that in our case, as an operator on the line Ran(P ) = {cv : c ∈ C}, a = h± (λ)−1 P , where h± (λ) := g ± (λ) + trace(P T P − P T QD0 QT P ). The trace is real-valued since v is real-valued and since the kernel of T is real-valued. In view of the definition of g ± (λ), h± (λ) = 0, provided λ > 0 is sufficiently small. Moreover, by (35) we see that A(λ)−1 = h± (λ)−1 S + QD0 Q, where S is of finite rank (in fact, the rank is at most two). By the definition of h± (λ) and by Lemma 8, |A(λ)−1 (·, ·)| + λ|∂λ A(λ)−1 (·, ·)| |S(·, ·)| + |QD0 Q(·, ·)|,
(36)
where the right-hand side is an L2 -bounded operator. Now M ± (λ)−1 = A(λ)−1 [1 + vE0± (λ)vA(λ)−1 ]−1 . The second inverse on the right-hand side exists for small λ since then 1 , 2 see (30). Moreover, writing out E ± (λ) as a Neuman series, one obtains (33) from (30) and (36) by termwise estimation. Finally, (34) is the well-known symmetric resolvent expansion which follows easily from vE0± (λ)vA(λ)−1 <
(I − U v(− + V − z)−1 v)(I + U v(− − z)−1 v) = I or V (− + V − z)−1 V = V − v(U + v(− − z)−1 v)−1 v for z > 0. Passing to the limit z → 0 now leads to (34) via an application of the resolvent identity and the limiting absorption principle, cf. (16).
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
101
Corollary 10. Let zero be a regular point of the spectrum of H = − + V . Then RV± (λ2 ) = R0± (λ2 ) − h± (λ)−1 R0± (λ2 )vSvR0± (λ2 ) − R0± (λ2 )vQD0 QvR0± (λ2 ) − R0± (λ2 )vE ± (λ)vR0± (λ2 ),
(37)
where S and E ± (λ) are as in the previous lemma. This is to be understood as an identity 1 1 between operators L2, 2 +ε (R2 ) → L2,− 2 −ε (R2 ) for small ε > 0, i.e., as in the limiting absorption principle (16). Proof. This is an immediate consequence of Corollary 6 and Lemma 9.
We now turn to decay estimates. Proposition 11. Let χ be a smooth cut-off function on the line with χ (λ) = 1 if λ ≤ λ1 and χ (λ) = 0 if λ ≥ 2λ1 , where λ1 > 0 is a small constant. Assume that zero is a regular point of the spectrum of H = − + V . Then √ 1 ∞ itλ2 |eitH χ ( H )Pac (H )f, g| = e λχ (λ) [RV+ (λ2 ) − RV− (λ2 )]f, g dλ π 0 ≤ C|t|−1 f 1 g1
(38)
for every f, g ∈ S(R2 ). Here C is a constant that only depends on V and χ . The proof of Proposition 11 is based on the expansion of RV± (λ2 ) stated in Corollary 10. Each of the four terms on the right-hand side of (37) requires a separate argument. We begin with the free case. Lemma 12. H0 = − satisfies |eitH0 χ ( H0 )Pac (H )f, g| ≤ C|t|−1 f 1 g1 for all f, g ∈ S(R2 ). itH0 f |t|−1 f and Proof. This follows ∞ 1 √ immediately from the standard bound e the fact that χ ( H0 ) and Pac (H ) are bounded on L1 (R2 ) (for the latter, use that the number of negative bound states is finite [Sto], as well as that the eigenfunctions are exponentially decaying by Agmon’s bound, and therefore in L1 (R2 ). Moreover, they are in L∞ (R2 ) by Sobolev imbedding). Alternatively, one can give a self-contained proof via stationary phase. Indeed, from (28),
R0+ (λ2 )(x, y) − R0− (λ2 )(x, y) =
i J0 (λ|x − y|). 2
Thus |eitH0 χ( H0 )Pac (H )f, g| 1 ∞ itλ2 ≤ e λχ (λ)J0 (λ|x − y|) dλ|Pac (H )f (x)||g(y)| dxdy. R4 2π 0
102
W. Schlag 1
Now J0 (u) = eiu ω+ (u) + e−iu ω− (u), where |ω± (u)| (1 + |u|)− 2 . Therefore, ∞ 2 eitλ λχ (λ)J0 (λ|x − y|) dλ 0 ∞ 2 ei[tλ −λ|x−y|] λχ (λ)ω+ (λ|x − y|) dλ (39) 0 ∞ 2 ei[tλ +λ|x−y|] λχ (λ)ω− (λ|x − y|) dλ. (40) + 0
Let t > 0. The phase in (39) has a stationary point λ0 = 1
|x−y| 2t .
Hence that integral is
1
t − 2 λ0 (1 + λ20 t)− 2 t −1 by stationary phase (we leave it to the reader to fill in the remaining details here). The integral in (40) can be estimated directly by means of integration by parts. The following lemmas deal with the contribution of the term containing QD0 Q in (37). In what follows it will be assumed that zero is a regular point of the spectrum of H = − + V . Lemma 13. Let (QD0 Q)(·, ·) denote the kernel of QD0 Q. There is the bound ∞ 2 eitλ λχ (λ)χ (λ|x − x1 |)Y0 (λ|x − x1 |)v(x1 )(QD0 Q)(x1 , y1 )v(y1 ) 8 R 0 J0 (λ|y1 − y|)χ (λ|y1 − y|) dλ f (x)g(y) dx1 dy1 dxdy ≤ C |t|−1 f 1 g1 (41) with a constant that only depends on V . Proof. We make the following claim: ∞
2 2 eitλ λχ (λ) χ (λ|x − x1 |)Y0 (λ|x − x1 |) − χ (λ(1 + |x|)) log(λ(1 + |x|)) π 0 −1 J0 (λ|y1 − y|)χ (λ|y1 − y|) dλ ≤ C |t| (1 + log+ |x1 | + log− |x − x1 |) (42) for all x, x1 , y, y1 ∈ R2 . Let k(x, x1 ) := 1 + log+ |x1 | + log− |x − x1 |.
(43)
If (42) holds, then the left-hand side of (41) is |t|−1 k(x, x1 )v(x1 )|(QD0 Q)(x1 , y1 )|v(y1 )|f (x)||g(y)| dx1 dy1 dxdy R8 1 1 2 k(x, x1 )2 |V |(x1 ) dx1 |QD0 Q| 2→2 V 12 f 1 g1 |t|−1 sup x∈R2
−1
|t|
R2
f 1 g1 ,
as desired. To see this, observe first that v(x)(QD0 Q)(x, y)h(y) dxdy = 0 R4
(44)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
103
for any h ∈ L2 (R2 ). Secondly, use Lemma 8 to control the L2 -operator norm of the kernel |QD0 Q|. To prove (42), let F (λ, x, x1 ) := χ (λ|x − x1 |)Y0 (λ|x − x1 |) − G(λ, y1 , y) := J0 (λ|y1 − y|)χ (λ|y1 − y|).
2 χ(λ(1 + |x|)) log(λ(1 + |x|)), (45) π (46)
If we choose 1 > λ1 > 0 so that 2λ1 lies to the left of the first zero of J0 , then G(λ, y1 , y) is nonincreasing in λ (recall the definition of χ in Proposition 11). Moreover, in that case 0 ≤ G ≤ 1 for all choices of arguments. Recall that J0 (z) = 1 + O(z2 ) and 2 (log z + c)J0 (z) + r(z), (47) π 2 2 2 Y0 (z) = J0 (z) + (log z + c)J0 (z) + r (z) = + g(z), (48) πz π πz where r(z) is analytic for all z and g(z) bounded on (0, ∞), say. Hence one has 1| F (0+, x, x1 ) = π2 c + π2 log |x−x 1+|x| , and G(0, y1 , y) = 1. It is easy to check that |x − x1 | (49) 1 + log+ |x1 | + log− |x − x1 | = k(x, x1 ). log 1 + |x| Y0 (z) =
Indeed, if |x| ≥ 2|x1 |, then 1 |x − x1 | 2|x| 1 χ[|x|≥1] + |x − x1 |χ[|x|≤1] ≤ ≤ ≤ 2. 4 2 1 + |x| 1 + |x| On the other hand, if |x| < 2|x1 |, then |x − x1 | 3|x1 | min(1, |x − x1 |) ≤ ≤ ≤ 3|x1 |, 1 + 2|x1 | 1 + |x| 1 + |x| and (49) follows. Integrating by parts inside the integral in (42) therefore leads to the estimate ∞ (42) |t|−1 k(x, x1 ) + |t|−1 |χ (λ)||F (λ, x, x1 )||G(λ, y1 , y)| dλ 0 ∞ −1 |∂λ F (λ, x, x1 )||G(λ, y1 , y)| dλ (50) +|t| 0 ∞ |F (λ, x, x1 )||∂λ G(λ, y1 , y)| dλ. (51) +|t|−1 0
Recall that the support of χ is contained inside [λ1 , 2λ1 ]. Thus the integral involving χ (λ) is easily seen to be sup |F (λ, x, x1 )||G(λ, y1 , y)| 1 + log− |x − x1 |, λ∼λ1
cf. (47). With the notation of (48),
2 1 χ (λ|x − x1 |) − χ (λ(1 + |x|)) ∂λ F (λ, x, x1 ) = πλ + |x − x1 |χ (λ|x − x1 |)Y0 (λ|x − x1 |) + |x − x1 |χ (λ|x − x1 |)g(λ|x − x1 |) 2 (52) − χ (λ(1 + |x|))(1 + |x|) log(λ(1 + |x|)). π
104
W. Schlag
Hence, (50) |t|−1 + |t|−1 −1
∞
0∞
0
+ |t|
∞
|χ (λ|x − x1 |) − χ (λ(1 + |x|))|λ−1 dλ |x − x1 |[|χ (λ|x − x1 |)| + χ (λ|x − x1 |)] dλ
(1 + |x|)|χ (λ(1 + |x|))| dλ
0
2|x − x | 2(1 + |x|)
1 |t|−1 1 + log+ + log+ |t|−1 k(x, x1 ), 1 + |x| |x − x1 | where we used (49) in the last step. In passing, we note that we have shown the following: 1 sup |F (λ, x, x1 )| ≤ |F (0, x, x1 )| + |∂λ F (λ, x, x1 )| dλ k(x, x1 ). (53) 0≤λ≤1
0
As observed previously, ∂λ G has a definite sign. Moreover, F (λ, x, x1 ) only has a finite number of zeros in λ. Hence, one can break up the integral (51) into finitely many disjoint intervals, remove the absolute values on each of them, and then integrate by parts. The only boundary contribution occurs at λ = 0, for which we have already obtained the desired bound. Otherwise, the remaining integral is bounded above by (50), and we are done. The following lemma deals with an integral very much like the one in (41). The difference here is that we consider the contribution from large arguments inside J0 , which makes it necessary to exploit the oscillations of J0 . This will be done by means of Lemma 2. Lemma 14. Let (QD0 Q)(·, ·) denote the kernel of QD0 Q. Let χ = 1 − χ. Then there is the bound ∞ 2 eitλ λχ (λ)χ (λ|x − x1 |)Y0 (λ|x − x1 |)v(x1 )(QD0 Q)(x1 , y1 )v(y1 ) 8 R 0 J0 (λ|y1 − y|) χ (λ|y1 − y|) dλ f (x)g(y) dx1 dy1 dxdy ≤ C |t|−1 f 1 g1 (54) with a constant that only depends on V . The same statement holds with the role of the cut-offs interchanged, i.e., with χ (λ|x − x1 |) and χ (λ|y − y1 |). Proof. As usual, J0 (y) = eiy ω+ (y) + e−iy ω− (y), 1
(55)
where |ω± (y)| (1 + |y|)− 2 − for all ≥ 0. Correspondingly, there will be two contributions to (54). We start with the phase φ− (λ) = λ2 − λ|y − y1 |t −1 which has a 1| critical point at λ0 = |y−y 2t . In that case we claim that ∞
2 eitφ− (λ) λχ (λ) χ (λ|x − x1 |)Y0 (λ|x − x1 |) − χ (λ(1 + |x|)) log(λ(1 + |x|)) π 0 ω− (λ|y1 − y|) χ (λ|y1 − y|) dλ ≤ C |t|−1 k(x, x1 ) (56) ( )
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
105
for all x, x1 , y, y1 ∈ R2 . Here k(x, x1 ) is as in (43). Moreover, as in the previous proof, this bound will lead to the desired estimate in (54) in view of (44). With F (λ, x, x1 ) as in (45), set χ (λ|y − y1 |)F (λ, x, x1 ), a(λ) := λχ (λ)ω− (λ|y − y1 |)
(57)
where we suppress the other variables inside a. By Lemma 2, ∞ ∞ |a (λ)| |a(λ)| itφ− (λ) −1 e a(λ) dλ |t| + χ[|λ−λ0 |>δ] dλ. 2 2 |λ − λ0 | 0 −∞ δ + |λ − λ0 | (58) To establish our claim we therefore need to show that the integral in (58) is k(x, x1 ). Using (53) one concludes ∞ |a(λ)| dλ 2 2 −∞ δ + |λ − λ0 | χ (λ|y − y1 |) λχ (λ)|ω− (λ|y − y1 |)| dλ k(x, x1 ) δ 2 + |λ − λ0 |2 √ 1 1 λ k(x, x1 )|y − y1 |− 2 dλ. (59) 2 2 c|y−y1 |−1 δ + |λ − λ0 | Now suppose first that λ0 δ, which is the same as |y − y1 |δ 1. Then √ 1 λ − 21 |y − y1 | dλ 2 + |λ − λ |2 −1 δ 0 c|y−y1 | √ 1 √ 1 λ0 |λ − λ0 | − 21 dλ + dλ |y − y1 | 2 2 δ 2 + |λ − λ0 |2 0 δ + |λ − λ0 | 0 1 1 |y − y1 |− 2 λ0 δ −1 + δ − 2 1 + (|y − y1 |δ)−1 1, as desired. On the other hand, if λ0 δ, then also |y − y1 |δ 1 and thus √ 1 λ − 21 |y − y1 | dλ 2 + |λ − λ |2 −1 δ 0 c|y−y1 | 1 1 3 λ− 2 dλ 1. |y − y1 |− 2 c|y−y1 |−1
It remains to bound the contribution of the term involving a (λ) in (58). Inspection of (52) reveals that |∂λ F (λ, x, x1 )| λ−1 . Combining this with (53) yields 1 (λ|y − y1 |) |a (λ)| k(x, x1 ) (χ (λ) + λ|χ (λ)|)(λ|y − y1 |)− 2 χ
(60) + χ (λ)|χ (λ|y − y1 |)| . We start with the second term in (60). Its contribution to the integral in (58) is 1 |χ (λ|y − y1 |)| dλ. χ[|λ−λ0 |>δ] |λ − λ0 | 0
(61)
106
W. Schlag
The integration region here is contained inside an interval of the form [c1 |y−y1 |−1 , c2 |y− y1 |−1 ], where c1 , c2 are some positive constants. If λ0 |y −y1 |−1 , then also |y −y1 |δ 1. Hence in this case (61) log(1 + δ −1 |y − y1 |−1 ) 1. If on the other hand either λ0 |y − y1 |−1 , or λ0 |y − y1 |−1 , then c |y − y |−1 − λ 2 1 0 (61) log 1. c1 |y − y1 |−1 − λ0 It remains to consider the first term in (60). Its contribution to the integral in (58) is dλ − 21 |y − y1 | χ[|λ−λ0 |>δ] (62) √ . |λ − λ0 | λ [λ|y−y1 |1] If λ0 |y − y1 |−1 , then 1
(62) |y − y1 |− 2
[λ|y−y1 |1]
dλ 3
λ2
1.
If, on the other hand, λ0 |y − y1 |−1 , then 1 21 λ0 dλ dλ − 21 (62) |y − y1 | χ[|λ−λ0 |>δ] √ + 3 1 λ0 λ 0 |λ − λ0 | 2 2 λ0 1
− 21
|y − y1 |− 2 (λ0
1
+ δ − 2 ) 1,
as desired. In the last line we used that λ0 |y − y1 |−1 is the same as |y − y1 |δ 1. This concludes the proof of claim (56). It remains to consider the phase φ+ (λ) = λ2 + t −1 |y − y1 |λ. The corresponding estimate is ∞
2 eitφ+ (λ) λχ (λ) χ (λ|x − x1 |)Y0 (λ|x − x1 |) − χ (λ(1 + |x|)) log(λ(1 + |x|)) π 0 ω+ (λ|y1 − y|) χ (λ|y1 − y|) dλ ≤ C |t|−1 k(x, x1 ) (63) for all x, x1 , y, y1 ∈ R2 . Setting a(λ) := λχ (λ)ω+ (λ|y − y1 ) χ (λ|y − y1 |)F (λ, x, x1 ), a single integration by parts in the left-hand side of (63) yields ∞ ∞
|a(λ)| |a (λ)| −1 (63) |t|−1 dλ + |t|
(λ)| dλ. 2 |φ+ (λ)| |φ+ 0 0 1| As before, λ0 = |y−y 2t . Then ∞ ∞ 1 |a(λ)| − 21 dλ k(x, x )|y − y | λ 2 (λ2 + λ20 )−1 χ[λ|y−y1 |1] dλ 1 1
2 |φ+ (λ)| 0 0 ∞ 3 1 λ− 2 |y − y1 |− 2 χ[λ|y−y1 |1] dλ k(x, x1 )
0
k(x, x1 ).
(64)
(65)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
107
To estimate the second integral in (65), we use (60) which remains valid with ω+ . Hence ∞ ∞
1 |a (λ)| − 21 λ− 2 (λ + λ0 )−1 χ[λ|y−y1 |1] dλ dλ k(x, x1 )|y − y1 |
|φ+ (λ)| 0 0 ∞ 3 1 λ− 2 |y − y1 |− 2 χ[λ|y−y1 |1] dλ k(x, x1 ) 0
k(x, x1 ). In view of the preceding, (65) |t|−1 k(x, x1 ). Hence (63) holds and (54) has been proved. The final statement about interchanging the roles of χ and χ is implicit in the previous proof. Indeed, (55) holds equally well for Y0 instead of J0 . Moreover, one replaces F (λ, x, x1 ) with G(λ, y, y1 ), see (46), and the bound (53) with the trivial one 0 ≤ G ≤ 1. We skip the details. The final lemma dealing with QD0 Q controls the contributions of those λ for which both resolvents on either side of vQD0 Qv are evaluated at arguments of size 1. In this case it will be convenient to work with the full kernel of the resolvents, i.e., the Hankel functions without splitting them into J0 and Y0 . Lemma 15. Let (QD0 Q)(·, ·) denote the kernel of QD0 Q and set χ = 1 − χ . There is the bound ∞ 2 eitλ λχ (λ) χ (λ|x − x1 |)H0± (λ|x − x1 |)v(x1 )(QD0 Q)(x1 , y1 )v(y1 ) 4 4 R 0 R H0± (λ|y1 − y|) χ (λ|y1 − y|) dλ f (x)g(y) dx1 dy1 dxdy ≤ C |t|−1 f 1 g1 (66) with a constant that only depends on V . Proof. One has H0+ (y) χ (y) = eiy ω+ (y) and H0− (y) χ (y) = e−iy ω− (y),
(67)
1
where ω− = ω+ , and |ω± (y)| (1 + |y|)− 2 − for all ≥ 0 (the reader should note that we are slightly abusing notation here, since ω± already appeared as the decay factors of J0 – but this abuse of notation is of no consequence). Correspondingly, there will be two phases to consider in (66), namely ( )
φ± (λ) = λ2 ± λ
|x − x1 | + |y − y1 | . t
Set p = |x − x1 | and q = |y − y1 | for simplicity. We may assume that p > 0 and q > 0. We claim that ∞ eitφ± (λ) λχ (λ) χ (λp)ω± (pλ) χ (λq)ω± (qλ) dλ |t|−1 , (68) 0
uniformly in p, q > 0. The phase φ− has a critical point at λ0 =
p+q . 2t
108
W. Schlag
Let a± (λ) = λχ (λ) χ (λp)ω± (pλ)ω± (qλ) χ (λq). Then by Lemma 2,
∞
e
itφ− (λ)
0
−1 a− (λ) dλ |t|
∞
|a− (λ)| dλ 2 + |λ − λ |2 δ 0 0 ∞
|a− (λ)| + |t|−1 χ[|λ−λ0 |>δ] dλ. |λ − λ0 | 0
(69)
The integral involving a− (λ) is
1
(pq)− 2
− 21
(pq)
1 c(p−1 +q −1 )
δ2
dλ + |λ − λ0 |2
δ −1 χ[λ0 p−1 +q −1 ] + (p −1 + q −1 )−1 1.
Here we used that λ0 p−1 + q −1 is the same as pq t or pqδ 2 1, as well as the bound √ pq − 21 −1 −1 −1 (pq) (p + q ) = 1. p+q Since 1
(λ)| (pq)− 2 λ−1 χ[λp−1 +q −1 ] χ (λ), |a−
(λ) in (69) is the integral involving a− 1
(pq)− 2
1 c(p−1 +q −1 )
χ[|λ−λ0 |>δ]
dλ . λ|λ − λ0 |
(70)
Now suppose that λ0 δ. Then |λ − λ0 | > δ implies that λ − λ0 λ. It follows that 1
(70) (pq)− 2
1 c(p−1 +q −1 )
1 dλ (pq)− 2 (p −1 + q −1 )−1 1. 2 λ
On the other hand, if λ0 δ which is the same as (p+q)δ 1, then by Cauchy-Schwarz − 21
(70) (pq)
− 21
(pq)
1
c(p−1 +q −1 )
(p
−1
+q
dλ 21 λ2
−1 − 21 − 21
)
δ
0
1
χ[|λ−λ0 |>δ]
dλ 21 |λ − λ0 |2
1
= (δ(p + q))− 2 1.
Hence (68) holds for the phase φ− . We now turn to φ+ . By inspection, 1
1
|a+ (λ)| (pq)− 2 χ[λp−1 +q −1 ] and |a+ (λ)| λ−1 (pq)− 2 χ[λp−1 +q −1 ] .
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
109
Integrating by parts therefore leads to ∞ ∞
|a+ (λ)| |a+ (λ)| −1 −1 (68) |t| dλ + |t|
(λ)| dλ 2 |φ+ (λ)| |φ+ 0 0 ∞ 1 (λ + λ0 )−2 χ[λp−1 +q −1 ] dλ |t|−1 (pq)− 2 0 ∞ − 21 −1 λ−1 (λ + λ0 )−1 χ[λp−1 +q −1 ] dλ + |t| (pq) 0 √ ∞ pq −1 − 21 −2 −1 |t| (pq) λ χ[λp−1 +q −1 ] dλ |t| |t|−1 , p + q 0 and thus (68) also holds for φ+ . We leave the remaining details to the reader.
(71)
We now combine Lemmas 13, 14, and 15 to obtain the following lemma. It bounds the contribution of the constant term in the expansion (32), see also (37). Lemma 16. For all test functions f, g and all t one has ∞ + 2 + 2 − 2 − 2 itλ2 e λχ (λ) R (λ )vQD QvR (λ ) − R (λ )vQD QvR (λ ) f, g dλ 0 0 0 0 0 0 0
|t|−1 f 1 g1
(72)
with a constant that only depends on V . Proof. Recall the representation (28) with H0± (z) = J0 (z) ± iY0 (z). Hence, R0+ (λ2 )(x, x1 )R0+ (λ2 )(y1 , y) − R0− (λ2 )(x, x1 )R0− (λ2 )(y1 , y) i = − (Y0 (λ|x − x1 |)J0 (λ|y − y1 |) + J0 (λ|x − x1 |)Y0 (λ|y − y1 |)). 8
(73)
In addition, we break up the integration region (0, ∞) by means of the partition 1 = χ(λ) + χ (λ). More precisely, write each resolvent as R0± (λ2 )(x, x1 ) = χ (λ|x − x1 |)R0± (λ2 )(x, x1 ) + χ (λ|x − x1 |)R0± (λ2 )(x, x1 ). This leads to four different terms in (72). Those terms that contain at least one χ (λ|x−x1 |) or χ (λ|y − y1 |) we rewrite further using (73). The other term which involves only χ we leave in terms of Hankel functions. Each of these different combinations is estimated by one of the previous three lemmas. Next we turn to the term involving S in (37). Lemma 17. Let S and h± (λ) be as in Lemma 9. Then for all test functions f, g and all t one has ∞ 1 1 + 2 + 2 − 2 − 2 itλ2 R R e λχ (λ) (λ )vSvR (λ ) − (λ )vSvR (λ ) f, g dλ 0 0 h (λ) 0 h (λ) 0 +
0
−1
|t|
f 1 g1
with a constant that only depends on V .
−
(74)
110
W. Schlag
Proof. Recall that S is of finite rank, and thus Hilbert-Schmidt. In particular, if S(x, y) denotes the kernel of S, then |S(x, y)| is again an L2 -bounded operator. Hence, one shows as before that (74) reduces to the bound
0
∞
2 eitλ λχ (λ) H0+ (λ|x − x1 |)H0+ (λ|y1 − y|)h−1 + (λ)
− H0− (λ|x − x1 |)H0− (λ|y1 − y|)h−1 − (λ) dλ
|t|−1 (1 + log− |x − x1 |)(1 + log− |y − y1 |).
(75)
As before, we set p := |x − x1 | and q := |y1 − y| for simplicity. We again need to distinguish whether or not the arguments of the Hankel functions are > 1 or < 1. This will be accomplished by means of the usual partition of unity 1 = χ + χ . It will also be important to remember that h+ (λ) = a log λ + z and h− (λ) = a log λ + z, where a = 0. It is understood that the cut-off χ (λ) in (74) is such that h± (λ) = 0 on the support of χ . One of the four terms in (75) which arises as a combination of χ and χ is
J0 (λp)J0 (λq) − Y0 (λp)Y0 (λq) dλ (log λ + c1 )2 + c22 0 ∞ [J0 (λp)Y0 (λq) + Y0 (λp)J0 (λq)](log λ + c1 ) 2 + eitλ λχ (λ)χ (λp)χ (λq) dλ (log λ + c1 )2 + c22 0 ∞
2
eitλ λχ (λ)χ (λp)χ (λq)
|t|−1 (1 + log− p)(1 + log− q).
(76) 2
2
1 d itλ This is proved by one integration by parts using λeitλ = 2it . In view of (47) the dλ e 4 fractions inside of the two integrals take the values π 2 and π4 , respectively, at λ = 0. Thus, the boundary terms contribute |t|−1 to the integration by parts. It remains to show that
∞
J0 (λp)J0 (λq) − Y0 (λp)Y0 (λq) d χ (λ)χ (λp)χ (λq) dλ dλ (log λ + c1 )2 + c22 0 ∞ [J0 (λp)Y0 (λq) + Y0 (λp)J0 (λq)](log λ + c1 ) d + χ (λ)χ (λp)χ (λq) dλ dλ (log λ + c1 )2 + c22 0 (1 + log− p)(1 + log− q).
If the derivative falls on χ (λ), then the resulting term is clearly bounded by (1 + log− p)(1 + log− q). On the other hand, suppose it falls on χ (λp). Then that term contributes
1 + log− (q/p) χ[p1] 1 + log− q, 1 + log+ p
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
111
and similarly if the derivative falls on χ (λq). It therefore remains to check that, with λ1 = cp −1 ∧ cq −1 ∧ c (c being some small constant)
d J (λp)J (λq) − Y (λp)Y (λq) 0 0 0 0 dλ 2 + c2 dλ (log λ + c ) 0 1 2 λ1
d [J0 (λp)Y0 (λq) + Y0 (λp)J0 (λq)](log λ + c1 ) + dλ dλ (log λ + c1 )2 + c22 0 λ1
(77) (78)
(1 + log− p)(1 + log− q).
We start with (77). Recall the expansion (48) for Y0 . Also, let n(λ) > 0 be such that n(λ)2 = (log λ + c1 )2 + c22 . Then clearly n(λ) ∼ | log λ| and n (λ) = λ−1 + O((λ log λ)−1 ) as λ → 0. Hence (77) 0
λ1
λ−1 + pg(λp) λ−1 + qg(λq) − (log (λq) + O(1)) + (log− (λp) + O(1)) n(λ)2 n(λ)2 (log− (λp) + O(1))(log− (λq) + O(1)) n (λ) dλ + 1. (79) +2 n(λ)3
1 Each of the three terms inside the absolute value contains an expression of the form λ log λ. Since these are not integrable, one needs to check that they cancel. Indeed, combining them yields
2 log λ 2(log λ)2
− n (λ) = O(λ−1 (log λ)−2 ), λn(λ)2 n(λ)3
(80)
which is integrable. Otherwise, we claim that (79) (1 + log− p)(1 + log− q). To see this, observe first that for all 0 < λ < λ1 , log− (λp) = log− λ + log− (p) = − log(λ) − log(p),
log− (λq) = − log(λ) − log(p).
Hence,
pg(pλ) − (log (λq) + O(1)) dλ n(λ)2 0 λ1 p|g(pλ)| (1 + log− (λ)) dλ(1 + | log q|) 2 n(λ) 0 p−1 p|g(pλ)| dλ(1 + log− q) λ1
(81)
0
1 + log− q. To pass to (81), note that if q ≥ 1, then sup 0<λ<λ1
1 + log− (λ) 1 + log− (λ)2 (1 + log q) sup 1, n(λ)2 n(λ)2 0<λ
(82)
112
W. Schlag
whereas if 0 < q < 1, then sup 0<λ<λ1
1 + log− (λ) (1 + log− q) 1 + log− q. n(λ)2
Furthermore,
λ1
0
λ−1 − (log (λq) − log λ + O(1))dλ n(λ)2 cq −1 ∧c
0
1 dλ (1 + | log q|) λ(log λ)2
1 (1 + | log q|) 1 + log− q. log− (cq −1 ∧ c)
(83)
Finally, we estimate
(log− (λp) + O(1))(log− (λq) + O(1)) − (log λ)2
n (λ) dλ n(λ)3 0 λ1
λ1 λ1 n (λ) log− (λq)
log− (λp)
dλ + n (λ) dλ + n (λ) dλ n(λ)3 n(λ)3 n(λ)3 0 0 0 λ1 log− (λp) log− (λq) − (log λ)2
+ n (λ) dλ . 3 n(λ) 0 λ1
(84) (85)
By our previous discussion, (84) 1 + log− q + log− p.
(86)
On the other hand,
| log p|| log− (λq)| + | log q| dλ λ(log λ)3 0 cp−1 ∧c cq −1 ∧cp−1 ∧c | log p| | log p|| log q| dλ + dλ 2 λ(log λ) λ(log λ)3 0 0 cq −1 ∧c | log q| + dλ λ(log λ)3 0 (1 + log− p)(1 + log− q).
(85)
λ1
(87)
Combining (87), (86), (82), (83) (and their analogues with p and q interchanged), as well as (80) yields that (79) (1 + log− p)(1 + log− q),
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
113
as claimed. As far as (78) is concerned, it will suffice to treat the term involving J0 (λp)Y0 (λq). This amounts to bounding
λ1 0
(λp 2 + O(λ3 p 4 ))(log− (λq) + O(1)) (log λ + c1 ) n(λ)2 (1 + O(λ2 p 2 ))(−λ−1 + qg(qλ)) + (log λ + c1 ) n(λ)2 (1 + O(λ2 p 2 ))(log− (λq) + O(1)) + λn(λ)2 2 2 (1 + O(λ p ))(log− (λq) + O(1))
−2 n (λ)(log λ + c ) dλ. 1 3 n(λ)
(88)
(89)
The first line (88) contributes 1 + log− q, as do all the O-terms in the other three lines. The remaining expression inside the absolute values is −2
n (λ) log λ + 2(log λ)2 = O(λ−1 (log λ)−2 ) 2 λn(λ) n(λ)3
as λ → 0. This establishes (76). Next we turn to the term containing the product χ (λp) χ (λq). In analogy with Lemma 15 we work with the Hankel functions rather than J0 , Y0 . Thus we need to show that ∞ λχ (λ) χ (λp) χ (λq)ω± (λq)ω± (λp) dλ |t|−1 ei[tλ±λ(p+q)] (90) h (λ) ± 0 uniformly in p, q > 0. Up to the factors h−1 ± this is the same as (68). Combine these factors with the λ-factor that appears in the integrand. This leads to functions that satisfy
d λ λ λ and 1 h± (λ) dλ h± (λ)
on the support of χ . Hence all the arguments from the proof of Lemma 15 apply to this case as well, and (90) holds. It remains to consider terms that contain χ (λp) χ (λq) or χ (λq) χ (λp). These terms are analogous to those in Lemma 14. We claim that
J0 (λp)J0 (λq) − Y0 (λp)Y0 (λq) dλ (log λ + c1 )2 + c22 0 ∞ [J0 (λp)Y0 (λq) + Y0 (λp)J0 (λq)](log λ + c1 ) 2 + eitλ λχ (λ) χ (λp)χ (λq) dλ (log λ + c1 )2 + c22 0 ∞
2
eitλ λχ (λ) χ (λp)χ (λq)
|t|−1 (1 + log− q). Write J0 , Y0 as J0 (y) = eiy ρ+ (y) + e−iy ρ− (y) and Y0 (y) = eiy σ+ (y) + e−iy σ− (y),
(91)
114
W. Schlag 1
where ρ± , σ± decay like y − 2 together with the natural derivative bounds. Thus (91) is the same as
ρ± (λp)J0 (λq) − σ± (λp)Y0 (λq) dλ (log λ + c1 )2 + c22 0 ∞ [ρ± (λp)Y0 (λq) + σ± (λp)J0 (λq)](log λ + c1 ) eitψ± (λ) λχ (λ) χ (λp)χ (λq) dλ + (log λ + c1 )2 + c22 0 ∞
eitψ± (λ) λχ (λ) χ (λp)χ (λq)
|t|−1 (1 + log− q),
(92)
where ψ± (λ) = λ2 ± pλ t . The bound (92) can be obtained by means of Lemma 2. In fact, the analysis in Lemma 14 carries over to this case with minor modifications. To see this, note that ρ± (λp)J0 (λq) − σ± (λp)Y0 (λq) χ (λp)χ (λq) λχ (λ) (log λ + c1 )2 + c22
(93)
1
λχ (λ) χ (λp)χ (λq)(λp)− 2 (1 + log− q) and also [ρ± (λp)Y0 (λq) + σ± (λp)J0 (λq)](log λ + c1 ) χ (λp)χ (λq) λχ (λ) (log λ + c1 )2 + c22
(94)
1
λχ (λ) χ (λp)χ (λq)(λp)− 2 (1 + log− q). And similarly for the derivatives. Since these bounds are the same (or even slightly better) than those satisfied by the functions a± in (57) and (64), the analysis of Lemma 14 pertaining to these functions carries over to this case as well, cf. (58), (63), and (65). This finishes the proof. In view of Corollary 10, the only remaining piece in the proof of Proposition 11 is that term in the expansion (37) which involves E ± . Lemma 18. Let E ± (λ) be as in Lemma 9. Then for all test functions f, g and all t one has
∞ 0
2 eitλ λχ (λ) R0+ (λ2 )vE + (λ)vR0+ (λ2 ) − R0− (λ2 )vE − (λ)vR0− (λ2 ) f, g dλ
|t|−1 f 1 g1 with a constant that only depends on V .
(95)
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
115
Proof. In analogy with Lemmas 13, 14, and 15 we divide the proof into three separate estimates namely, ∞ 2 eitλ λχ (λ)χ (λ|x − x1 |)H0± (λ|x − x1 |)v(x1 )E ± (λ)(x1 , y1 )v(y1 ) R8 0 ± H0 (λ|y1 − y|)χ (λ|y1 − y|) dλ f (x)g(y) dx1 dy1 dxdy ≤ C |t|−1 f 1 g1 , (96) ∞ 2 eitλ λχ (λ)χ (λ|x − x1 |)H0± (λ|x − x1 |)v(x1 )E ± (λ)(x1 , y1 )v(y1 ) R8 0 H0± (λ|y1 − y|) χ (λ|y1 − y|) dλ f (x)g(y) dx1 dy1 dxdy ≤ C |t|−1 f 1 g1 , (97) ∞ 2 eitλ λχ (λ) χ (λ|x − x1 |)H0± (λ|x − x1 |)v(x1 )E ± (λ)(x1 , y1 )v(y1 ) 4 4 R 0 R ± H0 (λ|y1 − y|) χ (λ|y1 − y|) dλ f (x)g(y) dx1 dy1 dxdy ≤ C |t|−1 f 1 g1 . (98) Unlike in the case of QD0 Q we do not exploit any cancellation between H0+ and H0− . This is not only impossible but also unnecessary. In contrast to QD0 Q, the logarithmic singularities of H0± at zero are compensated for by the vanishing of E ± (λ) at λ = 0, see (33). Let us start with that term where these singularities are not present, i.e., with (98). Set p = |x − x1 |, q = |y − y1 |, and λ0 = p+q 2t . Using the representation (67) and Lemma 2, we arrive at ∞ 2 ei[tλ −λ(p+q)] λχ (λ) χ (λp)ω− (λp)E − (λ)(x1 , y1 ) χ (λq)ω− (λq) dλ 0 ∞
(λ)| ∞ |a− |a− (λ)| |t|−1 dλ + χ dλ sup |E − (λ)(x1 , y1 )| [|λ−λ |>δ] 0 δ 2 + |λ − λ0 |2 |λ − λ0 | 0<λ<λ1 0 0 1 ∞ √ λ− 2 |a− (λ)| χ[|λ−λ0 |>δ] λ|∂λ E − (λ)(x1 , y1 )|, (99) dλ sup + |t|−1 |λ − λ | 0 0<λ<λ1 0 where we have set a− (λ) := λχ (λ) χ (λp)ω− (λp) χ (λq)ω− (λq). Note that the first two integrals involving a− appearing in (99) have already been treated in Lemma 15. Thus, the expression in braces is 1. Moreover, the third integral which 1
(λ), since the latter involves involves the new term λ− 2 a− (λ) is actually better than a− the loss of a full power of λ relative to a− rather than just a half power. Referring to the proof of Lemma 14 we can therefore again claim that the third integral in (99) is 1. All that remains now is to observe that (98) follows from the preceding by means of the error estimates (33). The case of E + is treated in an analogous fashion, see (71), and we skip the details. Next we consider the other extreme case, i.e., (96) in which H0± is only evaluated on the interval (0, 1]. Setting a± (λ) := χ (λ)χ (λp)ω± (λp)χ (λq)ω± (λq),
116
W. Schlag
a single integration by parts now yields ∞ 2 eitλ λχ (λ)χ (λp)ω± (λp)E ± (λ)(x1 , y1 )χ (λq)ω± (λq) dλ 0 ∞√ 1 −1
λ|a± (λ)| dλ sup λ− 2 |E ± (λ)(x1 , y1 )| |t| 0<λ<λ1
0
+ |t|−1
∞
λ
− 21
0
|a± (λ)| dλ
sup
√ λ|∂λ E ± (λ)(x1 , y1 )|.
(100)
0<λ<λ1
Now |a± (λ)| χ (λ)(1 + | log λ|2 )(1 + log− p)(1 + log− q),
|a± (λ)| χ[0<λ<1] λ−1 (1 + | log λ|)(1 + log− p)(1 + log− q).
To obtain (96), insert these bounds into (100) and invoke (33). It remains to consider the term of mixed type, i.e., (97). Thus set a− (λ) := λχ (λ) χ (λp)ω− (λp)χ (λq)ω− (λq). p one obtains Applying Lemma 2 with λ0 = 2t ∞ 2 ei[tλ −λp] λχ (λ) χ (λp)ω− (λp)E − (λ)(x1 , y1 )χ (λq)ω− (λq) dλ 0 ∞ √λ|a (λ)| − −1 |t| dλ 2 + |λ − λ |2 δ 0 0 √ ∞
(λ)| λ|a− 1 + χ[|λ−λ0 |>δ] dλ sup λ− 2 |E − (λ)(x1 , y1 )| |λ − λ0 | 0<λ<λ1 0 1 ∞ √ λ− 2 |a− (λ)| χ[|λ−λ0 |>δ] λ|∂λ E − (λ)(x1 , y1 )|. dλ sup + |t|−1 |λ − λ0 | 0<λ<λ1 0
(101)
The basic estimates on a− (λ) are 1
|a− (λ)| λχ (λ)(λp)− 2 (1 + log− λ)(1 + log− q), 1
|a− (λ)| χ[0<λ<1] (λp)− 2 (1 + log− λ)(1 + log− q).
Hence
√ 1 λ|a− (λ)| λχ (λ)(λp)− 2 (1 + log− q),
√
1 1 λ|a− (λ)| + λ− 2 |a− (λ)| χ[0<λ<1] (λp)− 2 (1 + log− q). These are precisely the bounds that were used in the proof of Lemma 14, and one can therefore repeat the arguments appearing there, see (58) to (62). Finally, the phase tλ2 + λp can be treated as in (65), and we skip the details. Acknowledgement. The author wishes to thank Monica Visan for comments on a preliminary version of this paper, as well as the anonymous referee for a very careful reading and many helpful comments.
Dispersive Estimates for Schr¨odinger Operators in Dimension Two
117
References [Agm]
Agmon, S.: Spectral properties of Schr¨odinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2, no. 2, 151–218 (1975) [GolSch] Goldberg, M., Schlag, W.: Dispersive estimates for Schr¨odinger operators in dimensions one and three. Commun. Math. Phys. 251, no. 1, 157–158 (2004) [JenNen] Jensen, A., Nenciu, G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13, no. 6, 717–754 (2001) [JenYaj] Jensen, A., Yajima, K.: A remark on Lp -boundedness of wave operators for two-dimensional Schr¨odinger operators. Commun. Math. Phys. 225, no. 3, 633–637 (2002) [JouSofSog] Journ´e, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schr¨odinger operators. Comm. Pure Appl. Math. 44, no. 5, 573–604 (1991) [Mur] Murata, M.: Asymptotic expansions in time for solutions of Schr¨odinger-type equations. J. Funct. Anal. 49 (1), 10–56 (1982) [RodSch] Rodnianski, I., Schlag, W.: Time decay for solutions of Schr¨odinger equations with rough and time-dependent potentials. Invent. Math. 155, 451–513 (2004) [Sto] Stoiciu, M.: An estimate for the number of bound states of the Schr¨odinger operator in two dimensions. Proc. Amer. Math. Soc. 132, no. 4, 1143–1151 (2004) . [Wed] Weder, R.: Lp -Lp estimates for the Schr¨odinger equation on the line and inverse scattering for the nonlinear Schr¨odinger equation with a potential. J. Funct. Anal. 170, no. 1, 37–68 (2000) [Yaj] Yajima, K.: Lp -boundedness of wave operators for two-dimensional Schr¨odinger operators. Commun. Math. Phys. 208, no. 1, 125–152 (1999)
Communicated by B. Simon
Commun. Math. Phys. 257, 119–149 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1323-8
Communications in
Mathematical Physics
Localization and Gluing of Topological Amplitudes Duiliu-Emanuel Diaconescu, Bogdan Florea Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA. E-mail: [email protected]; [email protected] Received: 6 May 2004 / Accepted: 25 October 2004 Published online: 22 March 2005 – © Springer-Verlag 2005
Abstract: We develop a gluing algorithm for Gromov-Witten invariants of toric Calabi-Yau threefolds based on localization and gluing graphs. The main building block of this algorithm is a generating function of cubic Hodge integrals of special form. We conjecture a precise relation between this generating function and the topological vertex at fractional framing. 1. Introduction A gluing algorithm for topological amplitudes on toric Calabi-Yau threefolds has been recently constructed in [2]. This algorithm is based on gluing topological vertices derived from large N duality and Chern-Simons theory. Previous work on the subject can be found in [1, 4–6, 11, 12]. In this paper we develop a parallel enumerative algorithm relying on localization and gluing of graphs. The main building block of this construction is a generating functional of cubic Hodge integrals, which is related to the topological vertex of [2]. The paper is structured as follows. Section Two is a review of local Gromov-Witten invariants associated to noncompact toric threefolds, localization and graphs. In Section Three we develop an algorithm for cutting and pasting of graphs from a pure combinatoric point of view. A concrete geometric implementation of this algorithm is presented in Sect. Four. The unit block can be formally written as a topological open string partition function for three lagrangian cycles in C3 . Applying open string localization [10, 13, 19], we obtain a generating function for cubic Hodge integrals. Section Five is devoted to a detailed comparison of this function with the topological vertex of [2]. We conjecture that the two expressions agree provided that the topological vertex is evaluated at fractional framing. A special case of this conjecture corresponding to a vertex with two trivial representations has been recently proved in [21, 22, 25]. We present strong numerical evidence for the general case by direct computations, but the proof is an open problem. Some technical details are included in two appendixes.
120
D.-E. Diaconescu, B. Florea
2. Localization and Graphs Let X be a smooth projective Calabi-Yau threefold. The Gromov-Witten invariants of X are defined in terms of intersection theory on the moduli space of stable maps M g,0 (X, β) with fixed homology class β ∈ H2 (X). More specifically, the moduli space M g,0 (X, β) has a special structure – perfect obstruction theory – which produces a virtual fundamental cycle of expected dimension [M g,0 (X, β)]vir ∈ A0 (M g,0 (X, β)). One defines the Gromov-Witten potential FX (gs , q) as a formal series FX (gs , q) =
∞
Cg,β q β ,
(2.1)
g=0 β∈H2 (X)
where Cg,β =
1.
(2.2)
[M g,0 (X,β)]vir
Here q β is a formal multisymbol satisfying q β+β = q β q β .
2.1. Local Gromov-Witten Invariants. In this paper we are mainly interested in noncompact toric Calabi-Yau threefolds X. The previous definition has to be refined since the moduli space M g,0 (X, β) is in principle ill-defined. Let X be a projective completion of X so that the divisor at infinity D = X \ X is reduced with normal crossings. Then there is a well-defined moduli space M g,0 (X, D, β) of relative stable maps to the pair (X, D) with multiplicity zero along D. Moreover, this moduli space has a well-defined perfect obstruction theory and a virtual fundamental cycle [M g,0 (X, D, β)]vir [17, 18]. For a class β ∈ H2 (X), this moduli space may contain closed connected components parameterizing maps supported away from D. We will denote the union of all these components by M g,0 (X, β). The virtual cycle [M g,0 (X, D, β)]vir induces a virtual cycle of expected dimension on M g,0 (X, β) by functoriality. Therefore we can define local Gromov-Witten invariants as in the compact case, taking into account the new meaning of M g,0 (X, β). Note that M g,0 (X, β) may be empty, in which case Cg,β = 0. To clarify this definition, let us consider some examples. First take X to be the total space of the canonical bundle KS over a toric Fano surface S, and let β be a curve class in the zero section. The completion can be taken to be X P(OS ⊕ OS (KS )). In this case M g,0 (X, β) M g,0 (S, β), and we could have adopted this as a definition of the moduli space. However, there are more general cases when such a direct approach is not possible. Consider for example the threefold defined by the toric diagram below. There are two compact divisors on X, S1 , S2 , both isomorphic to the Hirzebruch surface F1 , which intersect along a (−1, −1) curve. Any curve C lying on S1 ∪ S2 cannot be deformed in the normal directions, since both S1 , S2 are Fano. Therefore any map f : −→X with f∗ [] = [C] is supported away from the divisor at infinity. One could try to define local invariants in terms of maps to the singular divisor S1 ∪ S2 , but this approach would be quite involved. It is more convenient to use the construction explained in the previous paragraph, in which case the target space X is smooth.
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v1
v4
v2
v3
v5
v6 Fig. 1. A section in the toric fan of a local Calabi-Yau threefold X
2.2. Localization. Since X is toric, it admits a torus action T × X−→X which induces an action on the moduli space M g,0 (X, β). Then the local Gromov-Witten invariants can be computed by localization [9]. To recall the essential aspects, the virtual cycle [M g,0 (X, β)]vir induces a virtual cycle []vir on each component of the fixed locus. Moreover, one can construct a virtual normal bundle Nvir to each fixed locus. The localization formula reads 1 Cg,β = , (2.3) vir []vir eT (N ) ⊂M g,0 (X,β)
where eT denotes the equivariant Euler class. The fixed loci in the moduli space of stable maps can be indexed by graphs according to [14]. Since this construction will play an important role in the paper, let us recall the basic elements. Let {Pr }, r = 1, . . . , N denote the fixed points of the torus action on X. Any two fixed points are joined by a T -invariant rational curve Crs . The configuration of invariant curves forms a graph whose vertices are in 1 − 1 correspondence with the fixed points Pr and edges in 1 − 1 correspondence with curves Crs . Note that at most three edges can meet at any vertex. Some examples are represented below. The fixed maps f : −→X have a special structure. The image of f is contained in the configuration of invariant curves ∪r,s Crs . f −1 (Crs ) consists of finitely many irreducible components of which are smooth rational curves. The restriction of f to such a component must be a Galois cover. f −1 (Ps ) consists of finitely many prestable curves P2
P3
P4
P1 (a)
Fig. 2. The graph for a)
O(−3)−→P2
P5
P3
P6
P2
P1
(b) and b) the toric Calabi-Yau threefold represented in Fig. 1
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on possibly of higher genus. Note that higher genus components mapping onto some Crs are not allowed; all higher genus components must be collapsed to fixed points. To any irreducible component of the fixed locus we can associate a connected graph ϒ as follows. Let f : −→X denote a map in . i) The vertices v ∈ V (ϒ) represent prestable curves v ⊂ mapping to some fixed point Pi . Each vertex is marked by two numbers (kv , gv ), where kv ∈ {1, 2, . . . , N} is defined by f (v ) = Pkv , and gv is the arithmetic genus of v . Note that v may be a point. ii) The edges e ∈ E(ϒ) correspond to irreducible rational components of mapped onto Crs for some (r, s). Each edge is marked by an integer de representing the degree of the Galois cover f |e : e −→Crs . Let us define a flag [14] to be a pair (v, e) ∈ V (ϒ) × E(ϒ) so that v ∈ e. For a given v ∈ V (ϒ) we define the valence of v, val(v) to be the number of flags (v, e). We will also denote by F (ϒ) the set of flags of ϒ and by Fv (ϒ) the set of flags with given vertex v. Geometrically, val(v) counts the number of rational components e which intersect a given prestable curve v . Each flag determines a marked point p(v,e) ∈ v , so that (v , p(v,e) ) is a prestable curve of genus gv with val(v) marked points. According to [14], the set of all fixed loci is in one to one correspondence with (equivalence classes of) graphs ϒ subject to the following conditions: 1) If e ∈ E(ϒ) isan edge connecting two vertices u, v, then ku = kv . 2) 1 − χ (ϒ) + v∈V (ϒ) gv = g, where gv is the arithmetic genus of the component v , and χ (ϒ) is the Euler characteristic of ϒ, χ (ϒ) = |V (ϒ)| − |E(ϒ)|. 3) e∈E(ϒ) de f∗ [e ] = β. 4) For all v ∈ V (ϒ), (v , p(v,e) ) is a stable marked curve. Note that the last condition gives rise to some special cases, namely (gv , val(v)) = (0, 1), (0, 2). If (gv , val(v)) = (0, 1), v = pv is a smooth point of . If (gv , val(v)) = (0, 2), v = pv is a node of lying at the intersection of two components e1 (v) , e2 (v) . 1 Next, let us outline the computation of the local contribution []vir vir for a eT (N )
fixed component with associated graph ϒ. Given thestructure of an arbitrary fixed map, the fixed locus is isomorphic to a quotient of v∈V (ϒ) M gv ,val(v) by a finite group G(ϒ). The finite group admits a presentation 1−→ Z/de −→G(ϒ)−→Aut(ϒ)−→1, (2.4) e∈E(ϒ)
where Aut(ϒ) is the automorphism group of the graph. The main tool is the tangent obstruction complex of a map f : −→X, which encodes the local structure of the moduli space near the point (, f ). We have 0−→Aut()−→H 0 (, f ∗ TX )−→T1 −→Def()−→H 1 (, f ∗ TX )−→T2 −→0, (2.5) where T1 , T2 are the infinitesimal deformation and respectively obstruction space of a map (, f ). Aut(), Def() denote the infinitesimal automorphism and respectively deformation groups of the domain . Note that if (f, ) represents a point in , there is an induced T -action on the complex (2.5). According to [9], the fixed part of (2.5) under the torus action determines the virtual cycle []vir while the moving part determines the normal bundle Nvir . Moreover the
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induced virtual class coincides with the ordinary fundamental class of regarded as an 1 orbispace. The integrand vir can be computed in terms of the graph ϒ using the eT (N )
normalization exact sequence 0−→f ∗ TX −→ fe∗ TX ⊕ fv∗ TX −→ (TPkv X)val(v) −→0. (2.6) e∈E()
v∈V ()
v∈V ()
Note that the terms of this sequence form sheaves over the fixed locus which may not be in general locally free. For localization computations we only need the equivariant K-theory classes of these sheaves which will be denoted by [ ]. The associated long exact sequence of (2.6) reads 0 −→ H 0 (, f ∗ TX ) −→ H 0 (e , fe∗ TX ) ⊕ TPkv X−→ (TPkv X)val(v) e∈E()
∗
−→ H (, f TX )−→ 1
v∈V ()
e∈E()
H
1
v∈V ()
(e , fe∗ TX ) ⊕ H 1 (v , Ov ) ⊗ TPkv X−→0. v∈V () (2.7)
This yields [Nvir ] =
[H 0 (e , fe∗ TX )m ] − [H 1 (e , fe∗ TX )m ]
e∈E(ϒ)
−
[H 1 (v , Ov ) ⊗ TPkv X] + (val(v) − 1)[TPkv X]
(2.8)
v∈V (ϒ)
+ [Def()m ] − [Aut()m ]. The moving part of the automorphism group consists of holomorphic vector fields on the horizontal components e which vanish at the nodes of lying on e . We can write [H 0 (e , Te )m ] − [Tp(v,e) e ]. (2.9) [Aut()m ] = e∈E(ϒ)
(v,e)∈F (ϒ),val(v)≥2
The moving infinitesimal deformations of are deformations of the nodes lying at least on one edge component [Def()m ] = [Tp(v,e) e ⊗ Tp(v,e) v ] (v,e)∈F (ϒ),(gv ,val(v)) =(0,1),(0,2)
+
[Tpv e1 (v) ⊗ Tpv e2 (v) ].
(2.10)
(v,e)∈F (ϒ),(gv ,val(v))=(0,2)
Collecting the facts, it follows that the local contribution of the fixed locus can be written as 1 1 F (e) G(v) = vir |Aut(ϒ)| e∈E(ϒ) de []vir eT (N ) e∈E(ϒ) v∈V (ϒ),(gv ,val(v))=(0,2) × H (v), (2.11) v∈V (ϒ),(gv ,val(v)) =(0,1),(0,2) (M gv ,val(v) )T
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where eT (H 1 (e , fe∗ TX )m )eT (H 0 (e , Te )m ) , eT (H 0 (e , fe∗ TX )m ) eT (TPkv X)
, G(v) = eT (Tpv e1 (v) )eT (Tpv e2 (v) ) eT (Tpv e1 (v) ) + eT (Tpv e2 (v) ) F (e) =
eT (H 1 (v , Ov ) ⊗ TPkv X)
H (v) =
eT (E∨ v ⊗ TPkv X)
(v,e)∈Fv (ϒ)
eT (Tp(v,e) e )(eT (Tp(v,e) e ) + eT (Tp(v,e) v ))
(v,e)∈Fv (ϒ)
=
(2.12)
.
eT (Tp(v,e) e )(eT (Tp(v,e) e ) − ψp(v,e) )
In the last equation Ev is the Hodge bundle on the Deligne-Mumford moduli space M gv ,val(v) and ψp(v,e) are Mumford classes associated to the marked points {p(v,e) }. To conclude this section, note that the Gromov-Witten potential (2.1) can be written as a sum over marked graphs ϒ satisfying condition (1) above Eq. (2.4). For each such graph we define the genus g(ϒ) = 2 − χ (ϒ) + v∈V (ϒ) gv and the homology class β(ϒ) = e∈ϒ de fe∗ [e ]. Then we have FX (gs , q) =
ϒ
|Aut(ϒ)|
1
2g(ϒ)−2 β(ϒ)
e∈E(ϒ) de
C(ϒ)gs
q
.
(2.13)
Note that FX (gs , q) depends only on the marked graph , hence we can alternatively denote it by F (gs , q). We can further reformulate (2.13) by noting that the data kv , v ∈ V (ϒ) is equivalent to a map of graphs φ : ϒ−→. Therefore a marked graph ϒ can be alternatively , φ) where (ϒ ) obtained from ϒ by deleting the markings kv , thought of as a pair (ϒ and φ : ϒ −→ is a map of graphs. Condition (1) above (2.4) is replaced by ). (1’) φ(u) = φ(v) for any two distinct vertices u, v ∈ V (ϒ In the following we will use the notation (ϒ, φ) for such a pair. 3. Gluing Algorithm – Combinatorics Our goal is to find a gluing formula for the Gromov-Witten invariants of X based on a decomposition of the graph into smaller units. The main idea is to construct a suitable generating functional for each such unit so that the full Gromov-Witten potential can be obtained by gluing these local data. In this section we will discuss purely combinatoric aspects of this algorithm. A geometric realization will be presented in the next section. To review our setup, we are given a graph satisfying the following conditions: i) There are no edges starting and ending at the same vertex. ii) Any two distinct vertices are joined by at most one edge. iii) At most three edges can meet at any given vertex. We will denote the vertices of by P ∈ V () and the edges by C ∈ E(). To any such graph we attach a formal series of the form F (gs , q) =
(ϒ,φ)
1 2g(ϒ)−2 β(ϒ,φ) C(ϒ, φ)gs q |Aut(ϒ, φ)| e∈E(ϒ) de
(3.1)
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125
with coefficients C(ϒ, φ) ∈ KT , where we sum over (equivalence classes of) pairs (ϒ, φ) as above satisfying (1’).Here we define β(ϒ, φ) to be a formal linear combination of edges of , β(ϒ, φ) = e∈E(ϒ) de φ(e). q = (q1 , . . . , q|E()| ) is a multisymbol de associated to the edges of , and q β(ϒ,φ) = e∈E(ϒ) qφ(e) . We decompose into subgraphs, by specifying a collection of points Qα , α = 1, . . . , M lying on distinct edges C1 . . . , CM of . No two points should lie on the same edge. Suppose we choose these points so that is divided into several disconnected components I . The resulting graphs have more structure than the original graph . A typical graph I has two types of vertices: old vertices inherited from , and new univalent vertices resulting from the decomposition. We will also refer to old and new vertices as inner Vi () and respectively outer Vo () vertices. The edges of I can also be classified in inner edges Ei () – which do not contain outer vertices – and outer edges Eo () – which contain an outer vertex. Note that there is a unique outer edge CQ passing through each outer vertex Q. These graphs will be referred to as relative graphs. The decomposition of induces a similar decomposition of pairs (ϒ, φ). The points in the inverse image φ −1 ({Qα }) divide ϒ into disconnected graphs ϒI which map to I for each I . As before, a typical graph ϒI has more structure than the original graph ϒ. The decomposition gives rise to a collection of new univalent vertices in addition to the ordinary vertices inherited from ϒ. Moreover, the edges and ordinary vertices of ϒI inherit marking data from ϒ. The new vertices are unmarked. The new univalent vertices will be called outer vertices. The ordinary vertices will be referred to as inner vertices. An edge containing an outer vertex will be called outer edge. We denote by Vi,o (ϒI ), Ei,o (ϒI ) the set of inner/outer vertices and respectively edges. We also obtain a map of graphs φI : ϒI −→I which maps the distinguished vertices of ϒI to univalent vertices of I . To introduce some more terminology, we call the graphs ϒ closed graphs while ϒI will be called truncated graphs. Now, it is clear that all disconnected truncated graphs can be obtained by cutting closed graphs, and conversely, any closed graph can be obtained by gluing truncated graphs. We would like to use this idea in order to reconstruct the formal series (3.1) from data associated to the graphs I . For each I we need to construct a formal series with coefficients in KT by summing over equivalence classes of pairs (ϒI , φI ). In order to write down such an expression we need to introduce some more notation. Given a pair (ϒI , φI ) we define the genus g(ϒI ) = 1 − |Vi (ϒI )| + |Ei (ϒI )| +
(3.2)
gv ,
v∈Vi (ϒI )
and we denote by h(ϒI ) = |Vo (ϒI )| the number of outer vertices. For each univalent I , kI , . . . ) vertex of I , Q ∈ Vo (I ) we define a degree vector kIQ (ϒI , φI ) = (kQ,1 Q,2 I so that kQ,m is the number of outer edges of ϒI projecting onto the outer ray CQ I (ϒ , φ ) is an infinite vector with finitely many nonzero entries. with degree m. kQ I I Next, we have to introduce some formal variables keeping track of all this data. Let I I qI = (q1I , . . . , q|E ) and qI = ( q1I , . . . , q|E ) be associated to the inner and o ()| i (I )| respectively outer edges of . We define β(ϒI , φI ) =
e∈Ei (ϒI )
deI φI (e),
(ϒI , φI ) = β
e∈Eo (ϒI )
deJ φI (e)
(3.3)
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and β(ϒI ,φI )
qI
=
dI
(qφI I (e) ) e ,
(ϒI ,φI ) β
qI
=
e∈Ei (ϒI )
dI
( qφI I (e) ) e .
(3.4)
e∈Eo (ϒI )
I We also introduce formal variables yI = (yQ,m )m=1,... ,∞,Q∈Vo (I ) and set kI (ϒI ,φI )
yI
=
I kQ,m
I (yQ,m )
.
(3.5)
Q∈Vi (I ) m=1
Then the formal series associated to I takes the form ZI (gs , qI , q˜I , yI ) C(ϒI , φI ) 2g(ϒ )−2+h(ϒI ) β(ϒI ,φI ) w(ϒI ,φI ) kI (ϒI ,φI ) = gs I qI q˜I yI , I |Aut(ϒI , φI )| e∈E(ϒI ) de (ϒI ,φI )
(3.6) where the coefficients C(ϒI , φI ) ∈ KT . Note that here we sum over all disconnected marked graphs ϒI , as opposed to (3.1) where we sum over connected graphs. Now suppose we are given two relative graphs I , J . Choose a subset of outer vertices of I , SI ⊂ Vo (I ), and a subset SJ ⊂ Vo (J ) so that SI SJ . We glue I and J by choosing a bijection ψ : SI −→SJ , obtaining a relative graph I J with outer vertices Vo (I J ) = (Vo (I ) \ SI )∪(Vo (J ) \ SJ ) and outer edges Eo (I J ) = (Eo (I ) \ EI )∪ (Eo (J ) \ SJ ) (SI , SJ can be equally well regarded as subsets of Eo (I ), Eo (J ).) The inner vertices of I J are the union Vi (I J ) = Vi (I ) ∪ Vi (J ). The inner edges of I J are given by Ei (I J ) = Ei (I ) ∪ Ei (J ) ∪ S, where S denotes the set of inner edges of I J obtained by gluing outer edges of I , J ; S SI SJ . To I J we associate a series C(ϒI J , φI J ) 2g(ϒ )−2+h(ϒI J ) qI J , yI J ) = gs I J ZI J (gs , qI J , |Aut(ϒI J , φI J )| e∈E(ϒI J ) deI J (ϒI J ,φI J )
β(ϒI J ,φI J ) w(ϒI J ,φI J ) kI J (ϒI J ,φI J ) qI J yI J
×qI J
(3.7)
qI J , yI J are defined in terms of qI , qI , yI defined as above. The formal variables qI J , and qJ , qJ , yJ as follows: I qe , if e ∈ Ei (I ) qeI , if e ∈ Eo (I ) \ SI IJ IJ J qe = , qe = if e ∈ Ei (J ) , qe , J if e ∈ Eo (J ) \ SJ qe , qJ , if e ∈ S qI eI e if Q ∈ Vo (I ) \ SI y Q , IJ yQ = . (3.8) J yQ , if Q ∈ Vo (J ) \ SJ We would like to represent (3.7) as a pairing of the form qI J , yI J ) = ZI (gs , qI , qI , yI ), ZJ (gs , qJ , qJ , yJ ) , ZI J (gs , qI J ,
(3.9)
Localization and Gluing of Topological Amplitudes
P1
x Q
P2
127
P1
Q
ψ(Q)
γ1
γ
P2
γ2
Fig. 3. Decomposition of the graph associated to O(−1) ⊕ O(−1)−→P1
based on gluing of pairs (ϒI , φI ), (ϒJ , φJ ). Suppose (Q, ψ(Q)) ∈ SI × SJ are two I , CJ outer vertices of I , J identified in the gluing process. We denote by CQ ψ(Q) the corresponding outer edges of I , J . A pair of truncated graphs (ϒI , φI )(ϒJ , φJ ) can I match the be glued if and only if the degrees of all outer edges of ϒI projecting onto CQ J degrees of all outer edges of ϒJ projecting onto Cψ(Q) . Therefore two pairs (ϒI , φI ) and (ϒJ , φJ ) can be glued to form a pair (ϒI J , φI J ) if and only if I J kQ = kψ(Q)
∀ Q ∈ SI .
(3.10)
Note that if this condition is satisfied, one can identify any outer edge of ϒI projecting I to an outer edge of ϒ projecting to C J to CQ J ψ(Q) as long as the degrees are equal. This gives rise to (finitely) many different gluing combinations which may result in principle in different graphs ϒI J . In fact, is not hard to work out the degeneracy of each pair (ϒI J , φI J ) obtained by gluing a fixed pair (ϒI , φI ),(ϒJ , φJ ). By construction, we have a canonical embedding of groups Aut(ϒI J , φI J ) → Aut(ϒI , φI ) × Aut(ϒJ , φJ ).
(3.11)
Given a particular gluing of the pairs (ϒI , φI ),(ϒJ , φJ ) one can obtain another gluing compatible with (ϒI J , φI J ) by separately acting with elements of Aut(ϒI , φI ), Aut(ϒJ , φJ ) on each pair. Apparently this gives rise to |Aut(ϒI , φI )||Aut(ϒJ , φJ )| gluing patterns resulting in the same pair (ϒI J , φI J ). However, two of these patterns are equivalent if they are related by an element of Aut(ϒI J , φI J ) which acts simultaneously on the pairs (ϒI , φI ),(ϒJ , φJ ) through the embedding (3.11). Therefore the degeneracy of (ϒI J , φI J ) is |Aut(ϒI , φI )||Aut(ϒJ , φJ )| . |Aut(ϒI J , φI J )|
(3.12)
Moreover, since thenumber of allI possible gluing patterns of two fixed pairs (ϒI , φI ),(ϒJ , φJ ) is Q∈SI ∞ m=1 (kQ,m )! we have the following formula: ∞
I (kQ,m )! =
Q∈SI m=1
(ϒI J ,φI J )
|Aut(ϒI , φI )||Aut(ϒJ , φJ )| , |Aut(ϒI J , φI J )|
(3.13)
where the sum is over all pairs (ϒI J , φI J ) obtained by gluing (ϒI , φI ),(ϒJ , φJ ). Since this argument is perhaps too abstract, some concrete examples may be clarifying at this point. It suffices to consider a very simple situation in which is a graph with two vertices, which is the case for example if X is the total space of O(−1) ⊕ O(−1)−→P1 . We divide into two relative graphs by cutting the edge joining the two vertices as shown below.
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D.-E. Diaconescu, B. Florea 2
2
3
5
5
6
3
3
5
5
6
3
3
3
2
6
5
5
2 5
5 3
5 3
5
5
5
2
6
2
3
3
3
6
6 3
Fig. 4. First gluing example
We consider two examples of gluing graphs represented in Fig. 4 and Fig. 5 below. In both cases, we draw the pair (ϒI , φI ),(ϒJ , φJ ) on the top row and all possible gluing patterns resulting in graphs (ϒI J , φI J ) on the second row. ∞ 1 (ϒ , φ ) = (0, 1, 2, 0, 2, 1, 0, 0, . . . ), hence 1 In Fig. 4 we have kQ 1 1 m=1 (kQ,m )! = 2! × 2! = 4. Aut(ϒ1 , φ1 ) = {1}, Aut(ϒ2 , φ2 ) = {1}. There are four distinct gluing patterns, each resulting in a connected closed string graph with trivial automor1 phism group. Therefore formula (3.13) holds. For the pair in Fig. 5 we have kQ,m = ∞ 1 (0, 0, 2, 0, 0, 0, 3, 0, . . . ), hence m=1 (kQ,m )! = 2! × 3! = 12. Aut(ϒ1 , φ1 ) Aut(ϒ2 , φ2 ) S2 × S2 . There are two distinct gluing patterns resulting in disconnected graphs with automorphism groups S2 and respectively S2 ×S2 . Again the formula (3.13)
3
3
3
3
7
7
7
7
7
7
3
3
3
3
7
7
7
7
7
7
Fig. 5. Second gluing example
Localization and Gluing of Topological Amplitudes
129
holds. If the condition (3.10) is satisfied, one can easily show that 2g(ϒI J ) − 2 + h(ϒI J ) = 2g(ϒI ) − 2 + h(ϒI ) + 2g(ϒJ ) − 2 + h(ϒJ ), βI J (ϒI J , φI J ) = βI (ϒI , φI ) + βJ (ϒJ , φJ ) + de φI J (e), e∈S
I J (ϒI J , φI J ) = β I (ϒI , φI ) + β J (ϒJ , φJ ) β − de φI (e) − de φJ (e), IJ kQ (ϒI J , φI J ) =
e∈SI
(3.14)
e∈SJ
I (ϒ , φ ), kQ I I J kQ (ϒJ , φJ ),
if Q ∈ Vo (I ) \ SI if Q ∈ Vo (J ) \ SJ .
(3.15)
Using (3.8) and (3.9) we find that the relations (3.14) imply 2g(ϒI J )−2+h(ϒI J )
2g(ϒI )−2+h(ϒI ) 2g(ϒJ )−2+h(ϒJ ) gs , (ϒI J ,φI J ) β(ϒI J ,φI J ) β β(ϒI ,φI ) β(ϒI ,φI ) β(ϒJ ,φJ ) β(ϒJ ,φJ ) qI J qI J = qI qI qJ qJ .
gs
= gs
(3.16)
We define a formal pairing on y-variables by I k (ϒ ,φ ) kJ (ϒ ,φ ) yI I I , yJ J J = N (kSI I (ϒI , φI )) ×
∞
J kQ,m
J (yQ,m )
m=1
=
Q∈SI
∞
I kQ,m
I (yQ,m )
Q∈Vo (I )\SI m=1 ∞
Q∈Vo (J )\SJ
I I I J mkQ,m (kQ,m )! δ kQ,m , kψ(Q),m
m=1
k (ϒ ,φ ) N(kSI I (ϒI , φI ))yI JI J I J I J
Q∈SI
∞
m
I kQ,m
I I (kQ,m )! δ(kQ (ϒI , φI ),
m=1
J kψ(Q) (ϒJ , φJ )),
(3.17)
where N (kSI I (ϒI , φI )) is a phase factor depending only on the winding vectors of the I (ϒ , φ ) outer edges which take part in the gluing process kSI I (ϒI , φI ) = kQ . I i Q∈SI
The pairing is linear with respect to the other variables. The phase factor does not have a combinatoric explanation. It has to be included for geometric reasons explained in the next section. Using (3.16) and (3.17) we can compute the right-hand side of (3.9) ZI (gs , qI , qI , yI ), ZJ (gs , qJ , qJ , yJ ) = C(ϒI , φI )C(ϒJ , φJ ) (ϒI ,φI ) (ϒJ ,φJ ) (ϒ ,φ ) k (ϒ ,φ ) 2g(ϒI J )−2+h(ϒI J ) β(ϒI J ,φI J ) β qI J qI J I J I J yI JI J I J I J I kQ,m ∞ I (kQ,m )! Q∈SI m=1 m
×gs ×
|Aut(ϒI , φI )|
I |Aut(ϒ , φ )| J d d J J e∈E(ϒI ) e e∈E(ϒJ ) e
I J ×δ(kQ (ϒI , φI ), kψ(Q) (ϒJ , φJ )).
(3.18)
The δ-symbol in the right-hand side projects the sum onto pairs of graphs (ϒI , φI ), (ϒJ , φJ ) satisfying the gluing condition (3.10). In order for the right-hand side of (3.18)
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to agree with (3.7), the coefficients C(ϒI , φI ), C(ϒJ , φJ ) must satisfy the gluing condition C(ϒI , φI )C(ϒJ , φJ ) = N (kSI I (ϒI , φI ))∗ C(ϒI J , φI J )
(3.19)
for any pair (ϒI , φI ), (ϒJ , φJ ) satisfying (3.10), and for any pair (ϒI J , φI J ) obtained by gluing (ϒI , φI ), (ϒJ , φJ ). Obviously, this condition is not of a combinatoric nature. The coefficients in question must be specified by a particular geometric implementation of the gluing algorithm, which will be discussed in the next section. Here we will assume (3.19) to be satisfied, and show that the pairing (3.18) produces the expected result (3.7). Note that I kQ,m ∞ m Q∈SI m=1 1 = . (3.20) IJ I J e∈E(ϒI J ) de d d e∈E(ϒI ) e
e∈E(ϒJ ) e
I (ϒ , φ ), using the gluing condition (3.10). This follows from the definition of kQ I I Using (3.13), (3.19) and (3.20), in the right-hand side of (3.18) we find ZI (gs , qI , qI , yI ), ZJ (gs , qJ , qJ , yJ ) C(ϒI J , φI J ) = |Aut(ϒI J , φI J )| e∈E(ϒI J ) deI J (ϒI J ,φI J )
2g(ϒI J )−2+h(ϒI J ) β(ϒI J ,φI J ) w(ϒI J ,φI J ) kI J (ϒI J ,φI J ) qI J qI J yI J
×gs
(3.21)
which is the expected result (3.7). This is our main gluing formula. We would like to apply this gluing algorithm to the Gromov-Witten potential (3.1) which is a sum over connected graphs (ϒ, φ). One can construct a generating functional for disconnected graphs by taking the exponential of (2.13). It is a standard fact that Z (gs , q) = exp(F (gs , q)) can be written as a sum over disconnected graphs Z (gs , q) =
(ϒ,φ)
1 2g(ϒ)−2 β(ϒ,φ) C(ϒ, φ)gs q . |Aut(ϒ, φ)| e∈E(ϒ) de
(3.22)
One could use any decomposition of into relative graphs I . In particular we can cut along each edge, obtaining a collection of graphs P labeled by vertices P of . Each P has an inner vertex P and three outer vertices. These graphs will be simply called vertices. The main problem is finding a natural geometric construction for the coefficients C(ϒP , φP ) associated to P satisfying the gluing conditions (3.19). This is the subject of the next section. 4. Gluing Algorithm – Geometry This section consists of a geometric realization of the gluing algorithm. We consider a decomposition of induced by intersecting the invariant curves Crs with (noncompact) lagrangian cycles Lrs along circles Srs ; Srs divides Crs into two discs with common boundary. To each circle Srs we can associate a point Qrs on the corresponding edge of . The points Qrs divide into vertices as discussed in the last paragraph of the previous section. Each vertex represents a collection of (at most) three discs Di , i = 1, 2, 3 in C3 with common origin. The boundaries of the discs are contained in three lagrangian
Localization and Gluing of Topological Amplitudes
131
cycles Li , i = 1, 2, 3. Some vertices correspond to configurations of two or one discs, depending on the geometry. Those are special cases of the trivalent vertex. The main problem is finding a geometric construction for the generating functional (3.6) so that the coefficients C(ϒI , φI ) satisfy the gluing condition (3.19). A natural solution to this problem is suggested by string theory. One can wrap topological D-branes on the above lagrangian cycles, obtaining an open-closed topological string theory. Using the properties of this theory, one should be able to glue open string amplitudes obtaining closed string amplitudes. Therefore our generating functional should be the open string free energy associated to a collection of three lagrangian cycles in C3 as above. The main problem at this point is that there is no complete mathematical formalism for open string Gromov-Witten invariants. We can approach the problem from the point of view of large N duality and Chern-Simons theory as in [15, 26], or from an enumerative point of view as in [10, 13, 19, 20, 24]. The first approach has been implemented in [2], resulting in a gluing algorithm based on topological vertices. A topological vertex is the open string partition function of three lagrangian cycles in C3 as predicted by large N duality. Vertices can be naturally glued using a pairing very similar to (3.17). In this section we will take the second approach, constructing an open string generating functional based on heuristic localization computations as in [10, 13, 19, 24]. The resulting expression can be written as a sum over open string graphs, as explained below, therefore it is tailor made for our construction. We will compare it in detail to the topological vertex in the next section. Let us start with some basic facts. The open string Gromov-Witten invariants count virtual numbers of maps f : −→ C3 , f (∂) ⊂ L of fixed topological type, where is a genus g Riemann surface with h boundary components. The h boundary components are naturally divided into three groups, which are mapped to L1 , L2 and respectively L3 . We will denote by h1 , h2 , h3 , h1 + h2 + h3 = h the number of components in each group. We will also introduce three different sets of indices 1 ≤ ai ≤ hi , i = 1, 2, 3 in order to label the components in each group. The topological type of the map f is determined by three positive integers (d1 , d2 , d3 ) representing the degrees with respect to the three discs and three sets of winding numbers niai ≥ 0, ai = 1, . . . , hi , i = 1, 2, 3. In order to construct the generating functional, we introduce formal symbols q1 , q2 , q3 keeping track of the degrees and the formal variables zi = (zi,ai )ai =1,... ,∞ , i = 1, 2, 3 keeping track of the winding numbers, F (gs , qi , zi ) =
∞ ∞ g=0 h=1 di ,nia
2g−2+h
gs
Cg,hi (di |niai )
3
qidi
i=1
i
hi ai =1
nia
zi,aii .
(4.1)
Note that Cg,hi (di |niai ) = 0 unless di = haii =1 niai . This expression can be written in a more concise form if we introduce the winding vectors ki = (ki,m )m=1,... ,∞ . Each vector has finitely many nonzero entries which count the number of boundary components of mapping to each lagrangian cycle with given winding number. More precisely k i,m represents the number of boundary components mapping to Li with winding number m. ∞ Note that we have hi = ∞ k ≡ |k |, d = mk i,m i i i,m ≡ l(ki ). The coefficients m=1 m=1 Cg,hi (di |niai ) are invariant under permutations of boundary components mapping to the same cycle Li , hence they depend only on the ki . We can rewrite (4.1) as F (gs , qi , yi ) =
∞ g=0
2g−2
gs
ki
Cg (ki )
3 i=1
gs|ki | qi
l(ki )
3 i=1
yiki ,
(4.2)
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ki,m where yiki = ∞ m=1 yi,m . We have replaced the formal variables zi by new formal variables yi = (yi,m )m=1,... ,∞ which keep track of the winding vectors ki . So far open string Gromov-Witten invariants have been rigorously constructed for a single disc in C3 [20] equipped with a torus action. There is an alternative computational definition [13] based on a heuristic application of the localization theorem of [9] to open string maps. Although not entirely rigorous, the second approach has been tested in many physical situations with very good results [4–6, 10, 13, 19, 24]. We will apply the same technique in order to construct the generating functional (4.2). Given a circle action T × C3 −→ C3 preserving L, one can compute Cg,hi (di |niai ) by localization. The fixed open string maps can be labeled by graphs [10] by analogy with the closed string analysis of the previous section. The domain of a typical open string map is a union g,h = g0 ∪ ∪3i=1 ∪haii =1 iai where 0 is a closed prestable curve and iai are discs attached to g0 at the marked points pai i . The data (g0 , pai i ) must form a stable marked curve. The map f : −→ C3 collapses g0 to the origin P = {x1 = x2 = x3 = 0} and maps each iai to Di with degree niai . There are some special cases when g0 is a point, which have to be treated separately (see Appendix A.) Each fixed map is labeled by an open string graph with h rays attached to a single vertex v. The vertex represents 0 , hence it is marked by the arithmetic genus gv . The rays represent the discs iai , therefore they are marked by pairs (i, niai ). We will denote such marked graphs by . The generating functional (4.2) can be written as a sum over open string graphs F (gs , qi , yi ) =
1
2g( )−2+h( )
gs
|Aut( )|
3
i=1
hi
i ai =1 nai
C( )
3 i=1
l ( )
qi i
3
yiki ,
i=1
(4.3) where the notation is self-explanatory. For any graph we define g( ), h( ), ki ( ) to be the genus, number of rays and respectively the i th winding vector of the corresponding fixed map; li ( ) = l(ki ( )). The open string graphs are truncated graphs associated to the decomposition of in vertices, according to the terminology of the previous section. The data of the map φ is encoded in the markings of the rays. The sum over graphs (4.3) is the local potential (3.6) associated to a trivalent vertex. The coefficients C( ), or, equivalently, Cg,hi (di |niai ) are evaluated in Appendix A. In the remaining part of this section we will show that they satisfy the gluing conditions (3.19).
4.1. Gluing Conditions. Let us consider a pair of trivalent vertices r , s in the decomposition of which are glued to form a relative graph rs as in Fig. 6. The edge joining the two vertices corresponds to an invariant curve Crs on X. Let (−a, −2 + a), a ∈ Z denote the type of Crs . Consider two arbitrary open string graphs r , s projecting to r , s which satisfy the gluing condition κ1r = κ1s . Let rs be a new open string graph projecting to rs corresponding to an arbitrary gluing pattern of r , s . Here we want to prove the relation r ( ))C( )C( ), where N (kr ( )) is a phase factor. This is an C( rs ) = N(kQ r r s r Qr1 r1 essential condition for the gluing algorithm. Open string graphs can be evaluated using localization by analogy with closed string graphs. The open string coefficients have the following form:
...
...
... ...
...
...
...
...
...
...
Λ rs Q
Q r2
ψ Q r1
Q s1
γr
Q r2
s2
Pr
Ps
γs
Q
Q r3
Q s3
x Q rs
s2
Ps
γ rs
Q s3
...
Q r3
...
...
Λs
Λr
Pr
133
... ...
...
Localization and Gluing of Topological Amplitudes
Fig. 6. Gluing open string graphs
C( I ) =
eI ∈E( I )
×
FI (eI )
GI (vI )
vI ∈Vi ( I ),(gvI ,val(vI ))=(0,2)
Hr (vr ),
(4.4)
vI ∈Vi ( I ),(gvI ,val(vI )) =(0,1),(0,2) (M gvI ,val(vI ) )T
where the index I takes values I = r, s, rs, and FI (eI ), GI (vI ), HI (vI ) are edge and respectively vertex factors. The explicit expressions are computed in Appendix A. In order to simplify the notation, let us write C( r,s,rs ) = Ce ( r,s,rs )Cv ( r,s,rs ) separating the edge and the vertex factors. Note that the set of inner vertices of rs is Vi ( rs ) = Vi ( r )∪Vi ( s ). Moreover, the vertex factors Gr,s,rs (vr,s,rs ), Hr,s,rs (vr,s,rs ) are combinations of Hodge and Mumford classes determined by the marking data and the valence attached to a particular vertex. Since any inner vertex of rs comes from an inner vertex in either r or s , it follows that Cv ( r )Cv (λs ) = Cv ( rs ).
(4.5)
This leaves the edge factors. We have two types of edges. The outer edges associated to the univalent vertices Qr2 , Qr3 , Qs2 , Qs3 are preserved by the gluing together with their markings. Therefore the corresponding edge factors remain trivially unchanged. The interesting edges are those associated to the univalent vertices Qr1 , Qs1 which are identified in the gluing process. Geometrically, this corresponds to gluing two discs Dr1 , Ds1 along their boundaries, obtaining the smooth rational curve Crs . Before the gluing we have two products Ce1 ( r ), Ce1 ( s ) of open string edge factors. After the gluing we have a product Ce1 ( rs ) of closed string edge factors. All three products have the same number of factors, one for each edge of rs projecting to Crs . Therefore the proof of the gluing conditions reduces to proving that Fer ( r )Fes ( s ) = Fers ( rs )
(4.6)
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for any pair of edges er , es glued in the process. The edges er , es correspond to two T -fixed open string maps fr,s : r,s −→D1r,1s with the same degree dr = ds = d. The edge ers represents a T -fixed closed string map frs : rs P1 −→Crs of degree d. Recall that we denote by Lrs the lagrangian cycle which intersects Crs along the common boundary of D1r , D1s . A routine computation (see Appendix B) shows that Frs (ers ) = (−1)1+d(a−2) Fr (er )Fs (es ).
(4.7)
Therefore we conclude that ∞
C( r )C( s ) =
(−1)
r 1+m(a−2)kQ
r1 ,m
C( rs ).
(4.8)
m=1
This is the required gluing condition (3.19), in which the phase factor is a sign depending r ( ) = ks ( ). only on the winding vector kQ r s Qs1 r1 5. Topological Vertex: Localization versus Chern-Simons In this section we compare the open string free energy (4.1) with the topological vertex of [2]. The topological vertex is a generating functional for topological open string amplitudes derived from large N duality. Each lagrangian cycle Li carries a flat unitary gauge field Ai . We denote by Vi its holonomy around the boundary of the disc Di . Then the topological vertex is given by the following expression [2]: Z=
k1 ,k2 ,k3
Ckn11kn22 kn33
3 1 Tr i Vi , zk i k
(5.1)
i=1
∞ kj and Tr V = j kj and where ki are winding vectors, zk = k j kj !j j =1 (TrV ) n1 , n2 , n3 are the framing of the three legs of the vertex. The free energy derived from (5.1) is to be compared with the results from localization (see Appendix A.) For the computation of the necessary Hodge integrals we have used Faber’s Maple code [7] . Below we list the coefficients of several terms with h = 3, g ≤ 2 in the expansion of the free energy. i) (TrV1 )3 : Vertex result: ig 2 ig 3 2 n1 (n1 + 1)2 − n (n1 + 1)2 (8n21 + 8n1 − 9) 6 144 1
ig 5 2 n (n1 + 1)2 8n1 (n1 + 1)(13n21 + 13n1 − 34) + 189 . + 11520 1
(5.2)
Localization result: ig
ρ22 ρ32 6ρ14
− ig 3
ρ22 ρ32
144ρ16 +163ρ12 (ρ2 + ρ3 )2
ρ2ρ2 9ρ1 (ρ2 + ρ3 ) − 8ρ2 ρ3 + ig 5 2 3 8 − 26ρ13 (ρ2 + ρ3 ) 11520ρ1
− 272ρ1 ρ2 ρ3 (ρ2 + ρ3 ) + 104ρ22 ρ32 . (5.3)
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135
ii) (TrV1 )2 TrV2 : Vertex result: ig ig 3 (n1 + 1)2 − (n1 + 1)2 (2n21 + 4n1 + 1) 2 48
ig 5 (n1 + 1)2 4n1 (n1 + 2)(2n1 + 1) × (2n1 + 3) + 3 . + 11520
(5.4)
Localization result: ig
ρ32
ρ32
ρ13 + ρ12 (4ρ2 + ρ3 ) + 2ρ1 (2ρ22 + ρ2 ρ3 ) − 2ρ22 ρ3 48ρ14 ρ2 ρ2 ig 5 · 6 3 2 ρ16 + 2ρ15 × (19ρ2 + 5ρ3 ) + ρ14 (112ρ22 + 58ρ2 ρ3 + 5ρ34 ) + 11520 ρ1 ρ2 +4ρ13 ρ2 (41ρ22 + 24ρ2 ρ3 + 5ρ32 ) + 4ρ12 ρ22 (23ρ2 + 2ρ3 )
+8ρ1 ρ23 ρ3 (11ρ2 + 7ρ3 ) + 16ρ24 ρ32 . (5.5) 2ρ12
− ig 3
iii) (TrV1 )2 TrV22 : Vertex result:
ig 3 4 ig 2 n1 n2 + n1 (2n2 − 1) + 2n2 − 1) − n n2 + 2n31 (2n2 − 1) 2 24 1
+n21 (2n32 + 11n2 − 6) + 2n1 (2n32 − 3n22 + 7n2 − 3) + 4n32 − 6n22 + 6n2 − 2 ig 5 6 2n1 n2 + 6n51 (2n2 − 1) + 5n41 × (2n32 + 11n2 − 6) 1440 +20n31 (2n32 − 3n22 + 7n2 − 3) + n21 (6n52 + 110n32 − 180n22 + 183n2 − 60) +
+2n1 (6n52 − 15n42 + 70n32 − 90n22 + 59n2 − 15) + 6(2n52 − 5n42 + 10n32 − 10n22
+5n2 − 1) . (5.6) Localization result: ig
ρ32 (2ρ3 + ρ2 ) 2ρ12 ρ2 −ig 3
ρ32 (2ρ3 + ρ2 )
2ρ13 (ρ2 − ρ3 ) + 2ρ12 ρ2 (2ρ2 − ρ3 ) + 2ρ1 ρ23 − ρ23 ρ3 + ig 5 24ρ14 ρ23 ρ 2 (2ρ3 + ρ2 ) 6 4ρ1 (2ρ2 − 3ρ3 )(3ρ2 − ρ3 ) + 4ρ15 ρ2 (23ρ22 − 32ρ2 ρ3 + ρ32 ) × 3 2880ρ16 ρ25 +4ρ14 ρ22 (33ρ22 − 32ρ2 ρ3 − 6ρ32 ) + ρ13 ρ23 (87ρ22 − 73ρ2 ρ3 + 16ρ32 )
+ρ12 ρ24 (23ρ2 − 19ρ3 )(ρ2 − ρ3 ) − 2ρ1 ρ25 ρ3 × (11ρ2 + 3ρ3 ) + 4ρ26 ρ32 . (5.7) iv) TrV1 TrV2 TrV3 : Vertex result: ig −
ig 5 ig 3 + . 24 1920
(5.8)
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D.-E. Diaconescu, B. Florea
Localization result:
ig 3 2 ρ1 (ρ2 + ρ3 ) + ρ1 (ρ22 + 4ρ2 ρ3 + ρ32 ) + ρ2 ρ3 (ρ2 + ρ3 ) 24ρ1 ρ2 ρ3 ig 5 × 5ρ14 (ρ2 + ρ3 )2 + 2ρ13 (5ρ23 + 24ρ22 ρ3 + 24ρ2 ρ32 + 5ρ33 ) + 2 2 2 5760ρ1 ρ2 ρ3
ig −
+ρ12 (5ρ24 + 48ρ23 ρ3 + 102ρ22 ρ32 + 48ρ2 ρ33 + 5ρ34 )
+2ρ1 ρ2 ρ3 (5ρ23 + 24ρ22 ρ3 + 24ρ2 ρ32 + 5ρ33 ) + 5ρ22 ρ32 (ρ2 + ρ3 )2 .
(5.9)
v) TrV12 TrV2 TrV3 : Vertex result: ig(2n1 + 1) −
ig (2n1 + 1)(n21 + n1 + 1) 6
ig 5 (2n1 + 1) n1 (n1 + 1)(n21 + n1 + 3) + 1 . + 120
(5.10)
Localization result:
ρ1 + 2ρ2 3 ρ1 + 2ρ2 ρ1 (ρ2 + ρ3 ) + ρ12 (ρ22 + 6ρ2 ρ3 + ρ32 ) − 4ρ22 ρ32 − ig 3 ig 3 ρ1 24ρ1 ρ2 ρ3 ρ + 2ρ 1 2 +ig 5 × ρ16 (ρ2 + ρ3 )2 + 2ρ15 (ρ2 + ρ3 )(5ρ22 + 29ρ2 ρ3 + 5ρ32 ) 5 2 2 5760ρ1 ρ2 ρ3 +ρ14 (5ρ24 + 58ρ23 ρ3 + 186ρ22 ρ32 + 58ρ2 ρ33 + 5ρ34 ) − 40ρ13 ρ22 ρ32 (ρ2 + ρ3 )
−8ρ12 ρ22 ρ32 (9ρ22 + 46ρ2 ρ3 + 9ρ32 ) − 80ρ1 ρ23 ρ33 (ρ2 + ρ3 ) + 48ρ24 ρ34 . (5.11) vi) TrV12 TrV22 TrV3 : Vertex result: ig 3 3 4n1 (3n2 + 1) + 24n21 n2 + 3n1 (4n32 + 4n22 + 19n2 + 1) 24
ig 5 16n51 (3n2 + 1) + 160n41 n2 +2n2 × (4n22 + 15) + 1920 +40n31 (4n32 + 4n22 + 19n2 + 1) + 80n21 n2 × (4n22 + 15)
+n1 (48n52 + 80n42 + 760n32 +120n22 + 1167n2 + 5)+32n52 + 400n32 + 410n2 ) . (5.12)
ig(n1 + 2n2 + 3n1 n2 )−
Localization result: (2ρ2 + ρ1 )(2ρ3 + ρ2 ) 3 2 (2ρ2 + ρ1 )(2ρ3 + ρ2 ) − ig 3 ρ1 (ρ2 − 4ρ32 ) ig ρ1 ρ2 24ρ13 ρ23 ρ3
(2ρ2 + ρ1 )(2ρ3 + ρ2 ) +ρ12 ρ2 (ρ22 + 8ρ2 ρ3 − 4ρ32 ) − 4ρ1 ρ22 ρ32 − 4ρ23 ρ32 + ig 5 24ρ15 ρ25 ρ32 6 × ρ1 (5ρ24 − 72ρ22 ρ32 − 80ρ2 ρ33 + 48ρ34 ) + ρ15 ρ2 × (10ρ24 + 78ρ23 ρ3 − 80ρ22 ρ32 −576ρ2 ρ33 + 16ρ34 ) + ρ14 ρ22 (5ρ24 + 78ρ23 ρ3 + 240ρ22 ρ32 − 400ρ2 ρ33 +168ρ34 ) − 80ρ13 ρ23 ρ32 (ρ22 + 5ρ2 ρ3 − 4ρ32 ) − 8ρ12 ρ24 ρ32 (9ρ22 + 72ρ2 ρ3
−16ρ32 ) − 16ρ1 ρ25 ρ33 (5ρ2 − r3 ) + 48ρ26 ρ34 .
(5.13)
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137
vii) TrV12 TrV22 TrV32 : Vertex result:
ig 3 3 ig 2n1 (2n2 n3 + n2 + n3 ) + 2n2 n3 − 1 − 2n1 (2n2 n3 + n2 + n3 ) 6 + 3n21 (2n2 n3 − 1) + n1 (4n32 n3 + 2n32 + 6n22 n3 + 4n2 n33 + 6n2 n23 + 40n2 n3
+ 13n2 + 2n33 + 13n3 ) + 2n32 n3 − 3n22 + 2n2 n33 + 13n2 n3 − 3n23 − 4 ig 5 5 6n1 (2n2 n3 + n2 + n3 ) + 15n41 (2n2 n3 − 1) + 10n31 (4n32 n3 + 2n32 360 + 6n22 n3 + 4n2 n33 + 6n2 n23 + 40n2 n3 + 13n2 + 2n33 + 13n3 ) + 30n21 +
×(2n32 n3 − 3n22 + 2n2 n33 + 13n2 n3 − 3n23 − 4) + n1 (12n52 n3 + 6n52 + 30n42 n3 + 40n32 n33 + 60n32 n23 + 400n32 n3 + 130n32 + 60n22 n33 + 390n22 n3 + 12n2 n53 + 30n2 n43 + 400n2 n33 + 390n2 n23 + 1266n2 n3 + 299n2 + 6n53 + 130n33 + 299n3 ) + 6n52 n3 − 15n42 + 20n32 n33 + 130n32 n3 − 90n22 n23 − 120n22 + 6n2 n53
+ 130n2 n33 + 299n2 n3 − 15n43 − 120n23 − 48 . (5.14) Localization result: ig
(2ρ1 + ρ3 )(2ρ2 + ρ1 )(2ρ3 + ρ2 ) (2ρ1 + ρ3 )(2ρ2 + ρ1 )(2ρ3 + ρ2 ) + ig 3 ρ1 ρ2 ρ3 6ρ13 ρ23 ρ33 3 3 ρ1 (ρ2 + ρ22 ρ3 + ρ2 ρ32 + ρ33 ) + ρ12 (ρ23 ρ3 − 2ρ22 ρ32 + ρ2 ρ33 ) + ρ1 (ρ23 ρ32 + ρ22 ρ33 )
2ρ1 + ρ3 +ρ23 ρ33 + ig 5 × (2ρ2 + ρ1 )(2ρ3 + ρ2 ) ρ16 (ρ2 + ρ3 )2 (3ρ24 − 5ρ23 ρ3 5 5 5 360ρ1 ρ2 ρ3 + 15ρ22 ρ32 − 5ρ2 ρ33 + 3ρ34 ) + ρ15 ρ2 ρ3 × (ρ2 + ρ3 )(ρ24 − 41ρ23 ρ3 + 36ρ22 ρ32 − 41ρ2 ρ33 + ρ34 ) + ρ14 ρ22 ρ32 (8ρ24 − 5ρ23 ρ3 + 54ρ22 ρ32 − 5ρ2 ρ33 + 8ρ34 ) + 5ρ13 ρ23 ρ33 (ρ2 + ρ3 )(4ρ22 − 5ρ2 ρ3 + 4ρ32 ) + 8ρ12 ρ24 ρ34 (ρ22 − 5ρ2 ρ3 + ρ32 )
(5.15) + ρ1 ρ25 ρ35 (ρ2 + ρ3 ) + 3ρ26 ρ36 .
Using the condition ρ1 + ρ2 + ρ3 = 0 derived in Appendix A below, we find complete agreement between the two expansions provided that the framing variables ni are related to the torus weights by n1 =
ρ2 ρ3 ρ1 , n2 = , n3 = . ρ1 ρ2 ρ3
(5.16)
It is easy to check that there is no choice of the torus weights rendering all ni integral. This may seem puzzling at first since the framing variables are traditionally integral in Chern-Simons theory. A first deviation from this rule was noticed in [6] in the context of large N duality. Given the large N duality origin of the topological vertex, the present result is not surprising. As pointed out in [6], in order to obtain a consistent coupling of Chern-Simons theory and open string instanton corrections, the framing should be thought of as a formal variable. Then all Chern-Simons expressions must be formally expanded as a power series of these variables. The same strategy has been applied in this section, with very good results. We have also checked several terms with h = 4, g ≤ 2 and found agreement between the Chern-Simons and localization computations. In the
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D.-E. Diaconescu, B. Florea
light of this numerical evidence, we conjecture that the two generating functionals must agree to all orders. This result has been proved in [21, 22, 25] for a univalent vertex. The trivalent vertex is an open problem. Appendix A. Open String Localization Here we compute the generating functional (4.1) using open string localization. Let (x1 , x2 , x3 ) be coordinates on C3 , and let x1 −→e−iρ1 φ x1 ,
x2 −→e−iρ2 φ x2 ,
x3 −→e−iρ3 φ x3
(A.1)
be an S 1 action. The lagrangian cycles Li are defined by the following equations : L1 : L2 : L3 :
|x1 | = 1, |x2 | = 1, |x3 | = 1,
x2 = x 3 x 1 , x3 = x 1 x 2 , x1 = x 2 x 3 .
(A.2)
The S 1 action (A.1) preserves L = L1 ∪ L2 ∪ L3 if the weights (ρ1 , ρ2 , ρ3 ) satisfy ρ1 + ρ2 + ρ3 = 0.
(A.3)
The three S 1 -invariant discs ending on L are given by D1 : D2 : D3 :
0 ≤ |x1 | ≤ 1, 0 ≤ |x2 | ≤ 1, 0 ≤ |x3 | ≤ 1,
x2 = x3 = 0, x3 = x1 = 0, x1 = x2 = 0.
(A.4)
Let us describe the structure of an S 1 invariant map f : g,h −→X with lagrangian boundary conditions on L. The map f : g,h −→X is constrained by stability and S 1 invariance. We give a complete classification of all maps satisfying these two conditions, proceeding on a case by case basis. By S 1 invariance, f must map g,h onto the union of three discs D1 ∪ D2 ∪ D3 . In the generic case, the domain must be a nodal bordered Riemann surface, consisting of a closed surface g0 and three sets of discs iai , i = 1, 2, 3 which are mapped to D1 , D2 and respectively D3 . For future reference we will denote by tiai a coordinate on iai centered at the origin. The discs are attached to g0 by identifying the origins tiai = 0 to the marked points pai i ∈ g0 , so that we obtain a connected surface. The closed curve g0 is mapped to the common origin P of D1 , D2 , D3 . Stability further requires (g0 , pai i ) to be a stable marked curve. We obtain several cases which should be spelled out in detail. i) (g, h) = (0, 1). In this case, the domain is a single disc, which can be mapped to D1 , D2 or D3 . We have to distinguish accordingly three subcases: a) (g, hi ) = (0, 1, 0, 0), (di |niai ) = (d1 , 0, 0|d1 , 0, 0),
d1 f : 11 −→D1 , x1 = t11 ,
b) (g, hi ) = (0, 0, 1, 0), (di |niai ) = (0, d2 , 0|0, d2 , 0),
d2 f : 21 −→D2 , x2 = t21 ,
c) (g, hi ) = (0, 0, 0, 1), (di |niai ) = (0, 0, d3 |0, 0, d3 ),
d3 f : 31 −→D3 , x3 = t31 . (A.5)
The automorphism group is Aut(f ) Z/di , where i = 1, 2, 3.
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ii) (g, h) = (0, 2). The domain is a nodal (or pinched) annulus consisting of two discs with common origin. The two discs can be mapped either to the same disc Di in X or to two different discs Di , Dj , i = j . This yields again several subcases a) (g, hi ) = (0, 2, 0, 0), (di |niai ) = (d1 , 0, 0|n11 , n12 , 0, 0), f : 11 ∪ 12 −→D1 , n1
n1
n2
n2
n3
n3
x1 = t111 = t122 , 2 2 i b) (g, hi ) = (0, 0, 2, 0), (di |nai ) = (0, d2 , 0|0, n1 , n2 , 0), f : 21 ∪ 22 −→D2 , x2 = t211 = t222 , 3 3 i c) (g, hi ) = (0, 0, 0, 2), (di |nai ) = (d1 , 0, 0|0, 0, n1 , n2 ), f : 31 ∪ 32 −→D3 , x3 = t311 = t322 , (A.6) i d) (g, hi ) = (0, 1, 1, 0), (di |nai ) = (d1 , d2 , 0|d1 , d2 , 0), f : 11 ∪ 21 −→D1 ∪ D2 , d1 d2 x1 = t11 , x2 = t21 , e) (g, hi ) = (0, 1, 0, 1), (di |niai ) = (d1 , 0, d3 |d1 , 0, d3 ), f : 11 ∪ 31 −→D1 ∪ D3 , d1 d3 x1 = t11 , x3 = t31 , 2 3 i f ) (g, hi ) = (0, 0, 1, 1), (di |nai ) = (0, d2 , d3 |0, d2 , d3 ), f : 1 ∪ 1 −→D2 ∪ D3 , d2 d3 x2 = t21 , x3 = t31 . In the subcases (a), (b) and (c) the automorphism group is Zni × Zni , i = 1, 2, 3 for ni1 =
ni2 1 2 Aut(f ) = , Zni × Zni × Z/2, i = 1, 2, 3 for ni1 = ni2 1
(A.7)
2
where the Z/2 factor in the second line is generated by a permutation of the two components of the domain. This is an automorphism if and only if ni1 = ni2 . For the remaining three cases, the automorphism group is Aut(f ) = Z/di × Z/di+1
(A.8)
with the convention that 3 + 1 is identified with 1. Note that in this case, permuting the two components of the domain does not give rise to an automorphism even if di = di+1 for some i = 1, 2, 3. To conclude the classification of all possible fixed loci, we have one more case which has been briefly mentioned earlier, namely the generic case iii) (g, h) = (0, 1), (0, 2). The fixed map has the following form: f : g0 ∪ ∪ha11=1 1a1 ∪ ∪ha22=1 2a2 ∪ ∪ha33=1 1a3 −→D1 ∪ D2 ∪ D3 , (A.9) where f (g0 ) = P is a point, and n1
n1h
n2
n2h
n3
n3h 3
x1 = t111 = · · · = t1h11 , x2 = t211 = · · · = t2h22 ,
(A.10)
x2 = t311 = · · · = t3h3 . The marked Riemann surface (g0 , pai i ) must be a stable Deligne-Mumford curve. In this case the automorphism group is a product between G=
hi 3 i=1 ai =1
Z/niai
(A.11)
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D.-E. Diaconescu, B. Florea
and a subgroup P1 × P2 × P2 ⊂ Sh1 × Sh2 × Sh3 ,
(A.12)
where Pi permutes the marked points {pai i } preserving the winding numbers niai , i = 1, 2, 3. In terms of the winding vectors ki , we have Pi
∞
Ski,m .
(A.13)
m=1
Note that in all cases, the maps are fixed under the T action provided that T acts on the disc iai as follows: tiai −→e
−iφρi /nia
i
tiai .
(A.14)
The coefficients Cg,h1 ,h2 ,h3 (di |niai ) are computed by evaluating the contributions of the fixed points (i) − (iii) to the virtual fundamental class. As usual with open string localization computations, the result is a homogeneous rational function of (ρ1 , ρ2 , ρ3 ) of degree zero. In the cases (i) − (ii) above, the fixed locus in question is a point, therefore we have e(T2 ) 1 i Cg,h1 ,h2 ,h3 (di |nai ) = . (A.15) |Aut(f )| pt S 1 e(T1 ) Here e(T1,2 ) denote the equivariant Euler classes of the terms in the tangent obstruction complex restricted to the fixed locus, and the integral represents equivariant integration along the fibers the map pt S 1 −→BS 1 . For the third case, we have similarly e(T2 ) 1 Cg,h1 ,h2 ,h3 (di |niai ) = . (A.16) |Aut(f )| [M g,h ]S 1 e(T1 ) Next, we evaluate the contributions of the fixed points listed above starting with the generic case. Let f ∂ : g,h −→L denote the restriction of f : g,h −→X
to the boundary of g,h . The pair f ∗ TX , f∂∗ TL forms a Riemann-Hilbert bundle on g,h , ∂g,h and we will denote by TX the associated sheaf of germs of holomorphic sections. For future i the restriction of T to the disc i . For simplicity, we reference, we will denote by TXa X ai i will also denote the domain of f by , dropping the indices (g, h). The closed surface g0 will be similarly denoted by 0 , and the restriction f | 0 ≡ f 0 . The tangent-obstruction complex reads 0−→Aut()−→H 0 (, TX )−→T1 −→Def()−→H 1 (, TX )−→T2 −→0. (A.17) We denote the terms in this complex by B1 , . . . , B6 and the moving parts under the S 1 action by B1m , . . . , B6m . Then (A.16) becomes e(B1m )e(B5m ) 1 Cg,h1 ,h2 ,h3 (di |niai ) = . (A.18) |Aut(f )| [M g,h ]S 1 e(B2m )e(B4m ) We have e(B1m ) 1 = m e(B4 ) e(Def()m )
(A.19)
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since Aut() is generated in this case by tiai ∂tiai which are fixed by the S 1 action. The moving part of Def()m is generated by deformations of the nodes, that is Def()m ⊕3i=1 ⊕haii =1 Tpai 0 ⊗ T0 iai .
(A.20)
i
This yields e(Def() ) = m
hi 3 i=1 ai =1
ρi H − ψiai , niai
(A.21)
where H is the generator of H ∗ (BS 1 ) and ψiai ∈ H ∗ (M g,h ) are the Mumford classes associated to the marked points pai i for ai = 1, . . . , hi , i = 1, 2, 3. The other Euler classes in (A.16) can be evaluated using a (partial) normalization exact sequence i 0−→TX −→f0∗ TX ⊕ ⊕3i=1 ⊕haii =1 TXa −→ ⊕3i=1 ⊕haii =1 (TX )P −→0. i
(A.22)
The associated long exact sequence reads i ) 0 −→ H 0 (, TX )−→H 0 ( 0 , f0∗ TX ) ⊕ ⊕3i=1 ⊕haii =1 H 0 (iai , TXa i
−→ ⊕3i=1 ⊕haii =1 (TX )P i −→ H 1 (, TX )−→H 1 ( 0 , f0∗ TX ) ⊕ ⊕3i=1 ⊕haii =1 H 1 (iai , TXa )−→0. (A.23) i
We denote the terms in the complex (A.23) by F1 , . . . , F5 . Then we have e(F5m )e(F3m ) e(B5m ) = . m e(B2 ) e(F2m )
(A.24)
In principle we have all the elements needed for the evaluation of the r.h.s. of (A.24) i ). These groups can be computed as in except the cohomology groups H 0,1 (iai , TXa i [13, 19] or [5] obtaining the following expressions: i −1 n 1 a i i H 0 (iai , TXa ) (ρi ) ⊕ ρi ⊕ · · · ⊕ ρi ⊕ (0)IR , i niai niai niai − 1 1 2 1 i i H (ai , TXai ) ρi+1 + i ρi ⊕ ρi+1 + i ρi ⊕ · · · ⊕ ρi+1 + ρi , nai nai niai (A.25) where ρ3+1 should be identified with ρ1 . Now we can finish our Euler class computation e(F2m )
=H
d+3
3
(ρi )
di +1
i=1
e(F5m )
=H
d−h
3 i=1
hi i nai − 1 !
nia −1 , i i ai =1 nai ni −1
∗
ai hi
cg E (ρi H )
ai =1 l=1
l ρi+1 + i nai
(A.26)
,
(A.27)
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e(F3m ) = H 3h
3
ρih .
(A.28)
i=1
Collecting all intermediate results, we are left with e(B1m )e(B5m ) 1 Cg,h1 ,h2 ,h3 (di |niai ) = |Aut(f )| [M g,h ]S 1 e(B2m )e(B4m )
i 3 hi nai −1 i nai ρi+1 + lρi 1 (ρ1 ρ2 ρ3 )h−1 l=1 = (A.29) |P| ρ d1 ρ d2 ρ d3 (niai − 1)! 1 2 3 i=1 ai =1 H 2h−3 3i=1 cg (E∗ (ρi H )) × . 3 hi
i [M g,h ]S 1 i=1 ai =1 ρi H − nai ψiai
This represents the contribution of a generic fixed locus with (g, h) = (0, 1), (0, 2) to the virtual fundamental class. Next we evaluate the contributions of the fixed loci for the special cases (i) − (ii). In cases (ia) − (ic), the domain is a single disc i1 and f : i1 −→Di is a Galois cover of degree di . Using the same conventions and notations as above, we have di −1 e(B5m ) H −1 (di ρi+1 + lρi ) = , m d i e(B2 ) (di − 1)! ρi l=1
(A.30)
e(B1m ) ρi =H . m e(B4 ) di
(A.31)
The last equation follows from the fact that for a disc Def(i1 )m is trivial, while Aut(i1 )m is generated by ∂ti1 , which has weight ρdii . Taking into account the automorphism group, we obtain d1 −1 (d1 ρ2 + lρ1 ) 1 1 C0,1 (d1 , 0, 0|d1 , 0, 0) = 2 d −1 l=1 , 1 (d d1 ρ1 1 − 1)! d2 −1 (d2 ρ3 + lρ2 ) 1 1 C0,1 (0, d2 , 0|0, d2 , 0) = 2 d −1 l=1 , (A.32) 2 (d2 − 1)! d2 ρ2 d3 −1 (d3 ρ1 + lρ3 ) 1 1 C0,1 (0, 0, d3 |0, 0, d3 ) = 2 d −1 l=1 . (d3 − 1)! d3 ρ3 3 Next we consider case (ii). In the first three subcases (iia) − (iic) the domain of f is a nodal cylinder whose components are mapped in an invariant manner to the same disc in the target space X. It suffices to do the computations for (iia), since the remaining two cases are entirely analogous. We have to use a normalization sequence similar to (A.22), except that the closed curve g0 is absent. Therefore we have 1 2 0−→TX −→TX1 ⊕ TX1 −→(TX )P −→0
(A.33)
Localization and Gluing of Topological Amplitudes
143
which yields the following long exact sequence: 1 2 0 −→ H 0 (, TX )−→H 0 (11 , TX1 ) ⊕ H 0 (21 , TX1 )−→(TX )P −→H 1 (, TX ) 1 2 −→ H 1 (11 , TX1 ) ⊕ H 1 (21 , TX1 )−→0
(A.34)
whose terms will be denoted by F1 , . . . F5 as before. Then we can compute e(B5m ) e(F5m )e(F3m ) = e(B2m ) e(F2m ) n11 −1
= ρ 1 ρ2 ρ 3 H
l=1
n11 ρ2 + lρ1
n12 −1
1 l=1 n2 ρ2 ρ1d1 (n11 − 1)!(n12 − 1)!
+ lρ1
.
(A.35)
The remaining factors are e(B1m ) n11 n12 1 (ρ1 H )−1 = = e(Def()m ) e(B4m ) n11 + n12
(A.36)
since Aut()m is trivial, and Def()m is generated by deformations of the node Def()m T0 11 ⊗ T0 12 .
(A.37)
Substituting these expressions in (A.15), we obtain the following result: e(B1m )e(B5m ) 1 C0,2 (d1 , 0, 0|n11 , n12 , 0, 0) = |Aut(f )| pt S 1 e(B2m )e(B4m ) n12 −1 1 n11 −1 1 n1 ρ2 + lρ1 1 ρ1 ρ2 ρ3 l=1 l=1 n2 ρ2 + lρ1 . = |P| ρ d1 +1 (n11 − 1)!(n12 − 1)!(n11 + n12 ) 1
(A.38) The results for (iib) and (iic) can be obtained by permuting the weights and the winding numbers C0,2 (0, d2 , 0|0, n21 , n22 , 0)
1 ρ1 ρ2 ρ3 = |P| ρ d2 +1
n21 −1
n22 −1
l=1
l=1
n21 ρ3 + lρ2
n22 ρ3 + lρ2
, 2 − 1)!(n2 − 1)!(n2 + n2 ) (n 1 2 1 2 2 n32 −1 3 n31 −1 3 n ρ + lρ ρ ρ 1 ρ 1 3 1 2 3 1 l=1 l=1 n2 ρ1 + lρ3 3 3 . C0,2 (0, 0, d3 |0, 0, n1 , n2 ) = |P| ρ d3 +1 (n31 − 1)!(n32 − 1)!(n31 + n32 ) 3 (A.39)
This leaves us with subcases (iie) − (iif ). Again, it suffices to do the computations only for (iie). We have a map f : 11 ∪ 21 −→D1 ∪ D2 which is a Galois cover of D1 and respectively D2 when restricted to the components 11 , 21 . In this case the normalization exact sequence is 1 2 0−→TX −→TX1 ⊕ TX1 −→(TX )P −→0.
(A.40)
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D.-E. Diaconescu, B. Florea
The associated long exact sequence reads 1 2 ) ⊕ H 0 (21 , TX1 )−→(TX )P −→H 1 (, TX ) 0 −→ H 0 (, TX )−→H 0 (11 , TX1 1 2 −→ H 1 (11 , TX1 ) ⊕ H 1 (21 , TX1 )−→0.
Repeating the previous steps, this yields d1 −1 2 −1 e(B5m ) (d1 ρ2 + lρ1 ) dl=1 (d2 ρ3 + lρ2 ) ρ1 ρ2 ρ3 l=1 =H d d . m 1 2 e(B2 ) (d1 − 1)!(d2 − 1)! ρ ρ 1
(A.41)
(A.42)
2
Moreover, the moving part of Def() is generated again by deformations of the node Def()m T0 (11 ) ⊗ T0 (21 ),
(A.43)
e(B1m ) d 1 d2 H −1 . = m e(B4 ) d2 ρ1 + d 1 ρ2
(A.44)
which yields
Collecting all results we obtain the following expression : C0,2 (d1 , d2 , 0|d1 , d2 , 0) =
ρ1 ρ2 ρ3
d1 −1 l=1
(d2 ρ1 + d1 ρ2 )ρ1d1 ρ2d2
2 −1 (d1 ρ2 + lρ1 ) dl=1 (d2 ρ3 + lρ2 ) . (A.45) (d1 − 1)!(d2 − 1)!
For the remaining two cases, we can obtain the result by permuting the weights in (A.45) C0,2 (0, d2 , d3 |0, d2 , d3 ) =
ρ1 ρ2 ρ3
=
ρ1 ρ2 ρ3
(d3 ρ2 + d2 ρ3 )ρ2d2 ρ3d3 C0,2 (d1 , 0, d3 |d1 , 0, d3 )
d2 −1 l=1
d3 −1 l=1
(d1 ρ3 + d3 ρ1 )ρ3d3 ρ1d1
3 −1 (d2 ρ3 + lρ2 ) dl=1 (d3 ρ1 + lρ3 ) , (d2 − 1)!(d3 − 1)! 1 −1 (d3 ρ1 + lρ3 ) dl=1 (d1 ρ2 + lρ1 ) . (A.46) (d3 − 1)!(d1 − 1)!
To summarize this subsection, let us collect the results for Cg,hi (di |niai ), C0,1 (d1 , 0, 0|d1 , 0, 0) =
1 1 d12 ρ1d1 −1
1 1 C0,1 (0, d2 , 0|0, d2 , 0) = 2 d −1 d2 ρ2 2
d1 −1
(d1 ρ2 + lρ1 ) , (d1 − 1)!
l=1
d2 −1
(d2 ρ3 + lρ2 ) , (d2 − 1)!
l=1
d3 −1
(d3 ρ1 + lρ3 ) , (d3 − 1)! n12 −1 1 n11 −1 1 n1 ρ2 + lρ1 1 ρ1 ρ2 ρ3 l=1 l=1 n2 ρ2 + lρ1 1 1 , C0,2 (d1 , 0, 0|n1 , n2 , 0, 0) = 1 − 1)!(n1 − 1)!(n1 + n1 ) |P| ρ d1 +1 (n 1 2 1 2 1
C0,1 (0, 0, d3 |0, 0, d3 ) =
1 1 d32 ρ3d3 −1
l=1
Localization and Gluing of Topological Amplitudes
C0,2 (0, d2 , 0|0, n21 , n22 , 0)
1 ρ1 ρ2 ρ3 = |P| ρ d2 +1
145
n21 −1
n22 −1
l=1
l=1
n21 ρ3 + lρ2
n22 ρ3 + lρ2
, (n21 − 1)!(n22 − 1)!(n21 + n22 ) n32 −1 3 n31 −1 3 n1 ρ1 + lρ3 1 ρ1 ρ2 ρ3 l=1 l=1 n2 ρ1 + lρ3 3 3 , C0,2 (0, 0, d3 |0, 0, n1 , n2 ) = |P| ρ d3 +1 (n31 − 1)!(n32 − 1)!(n31 + n32 ) 3 2
C0,2 (d1 , d2 , 0|d1 , d2 , 0) =
ρ1 ρ2 ρ3
=
ρ1 ρ2 ρ3
=
ρ1 ρ2 ρ3
d1 −1 l=1
(d2 ρ1 + d1 ρ2 )ρ1d1 ρ2d2 C0,2 (0, d2 , d3 |0, d2 , d3 )
d2 −1 l=1
(d3 λ2 + d2 ρ3 )ρ2d2 ρ3d3 C0,2 (d1 , 0, d3 |d1 , 0, d3 )
d3 −1 l=1
(d1 ρ3 + d3 ρ1 )ρ3d3 ρ1d1
2 −1 (d1 ρ2 + lρ1 ) dl=1 (d2 ρ3 + lρ2 ) , (d1 − 1)!(d2 − 1)! 3 −1 (d2 ρ3 + lρ2 ) dl=1 (d3 ρ1 + lρ3 ) , (d2 − 1)!(d3 − 1)! 1 −1 (d3 ρ1 + lρ3 ) dl=1 (d1 ρ2 + lρ1 ) , (d3 − 1)!(d1 − 1)!
i hi nai −1 i 3 h−1 nai ρi+1 + lρi ρ ρ ) 1 (ρ 1 2 3 l=1 i Cg,h1 ,h2 ,h3 (di |nai ) = (niai − 1)! |P| ρ d1 ρ d2 ρ d3 1 2 3 i=1 ai =1 H 2h−3 3i=1 cg (E∗ (ρi H )) × . 3 hi
i [M g,h ]S 1 i=1 ai =1 ρi H − nai ψiai
(A.47)
Appendix B. The Gluing Condition for Open String Graphs In this appendix we prove the gluing formula (4.6) for an arbitrary invariant curve Crs of type (−a, −2 + a). Let U be an open neighborhood of Crs in X which can be covered by two smooth coordinate patches Ur , Us with coordinates (x1 , x2 , x3 ) and (y1 , y2 , y3 ). The local coordinates are chosen so that (x1 , y1 ) are affine coordinates on P1 , while (x2 , x3 ) and respectively (y2 , y3 ) are normal coordinates in the two patches. Therefore the transition functions are y1 =
1 , x1
y3 = x1−2+a x3 .
y2 = x1a x2 ,
(B.1)
In terms of local coordinates, the torus action reads x1 −→e−iρ1 φ x1 , x2 −→e−iρ2 φ x2 , y1 −→eiρ1 φ y1 , y2 −→e−i(aρ1 +ρ2 )φ y2 ,
x3 −→e−iρ3 φ x3 , y3 −→e−i((−2+a)ρ1 +ρ3 )φ y3 . (B.2)
Note that the local form of the action in the patch U2 is determined by the local action in U1 and the transitions functions (B.1). We denote by Pr :
x1 = x2 = x3 = 0,
the fixed points of the torus action.
Ps :
y1 = y2 = y3 = 0
(B.3)
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D.-E. Diaconescu, B. Florea
There are five lagrangian cycles in Urs = Ur ∪ Us given by Lr1 : Lr2 : Lr3 :
|x1 | = 1, |x2 | = 1, |x3 | = 1,
x2 = x 3 x 1 , x3 = x 1 x 2 , x1 = x 2 x 3 ,
Ls1 : Ls2 : Ls3 :
|y1 | = 1, |y2 | = 1, |y3 | = 1,
y2 = y 3 y 1 , y3 = y 1 y 2 , y1 = y 2 y 3 .
(B.4)
Note however that Lr1 = Ls1 are identical cycles. This can be seen using the transition functions (B.1). We also have six discs ending on the lagrangian cycles. For the present purposes we will consider only two of them Dr1 :
0 ≤ |x1 | ≤ 1,
x2 = x3 = 0,
Ds1 :
0 ≤ |y1 | ≤ 1,
y2 = y3 = 0.
(B.5)
We consider open string fixed maps fr : r −→Dr1 ,
fs : s −→Ds1
(B.6)
of the same degree n which yield a degree n closed string map frs : rs −→Crs
(B.7)
by gluing. We denote by TXr , TXs the corresponding Riemann-Hilbert bundles on r , s as in Appendix A. The edge factors are e(H 1 (r , TXr )m )e(Aut(r )m ) , e(H 0 (r , TXr )m ) e(H 1 (s , TXs )m )e(Aut(s )m ) Fs (es ) = , e(H 0 (s , TXs )m ) e(H 1 (rs , frs∗ TX )m )e(Aut(rs )m ) Frs (ers ) = . e(H 0 (rs , f ∗ TX )m )
Fr (er ) =
(B.8)
Let us compute the equivariant Euler classes in (B.8). For discs we can copy the results of the previous section (Eq.(A.25) ) taking into account the local form of the S 1 action (B.2) , n−1 e(H 1 (r , TXr )m ) −1 k=1 (nρ2 + kρ1 ) , =H ρ1n (n − 1)! e(H 0 (r , TXr )m ) (B.9) n−1 + ρ ) − kρ (n(aρ ) e(H 1 (s , TXs )m ) 1 2 1 = H −1 k=1 . e(H 0 (s , TXs )m ) (−ρ1 )n (n − 1)! In order to compute the edge factor for frs : rs −→X we have to use the short exact sequence of the image 0−→TCrs −→TX |Crs −→NCrs /X −→0.
(B.10)
This induces a short exact sequence on the domain1 0−→frs∗ TCrs −→frs∗ TX −→frs∗ NCrs /X −→0. 1
(B.11)
In order for the first and last term to make sense, we have to think of frs as a map to Crs instead of X. This is a slight abuse of notation.
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147
The associated long exact sequence reads 0 −→ H 0 (rs , frs∗ TCrs )−→H 0 (rs , frs∗ TX )−→H 0 (rs , frs∗ NCrs /X )−→ −→ H 1 (rs , frs∗ TCrs )−→H 1 (rs , frs∗ TX )−→H 1 (rs , frs∗ NCrs /X )−→0. (B.12) which shows that e(H 1 (rs , frs∗ TCrs )m )e(H 1 (rs , frs∗ NCrs /X )m ) e(H 1 (rs , frs∗ TX )m ) = . e(H 0 (rs , frs∗ TX )m ) e(H 0 (rs , frs∗ TCrs )m )e(H 0 (rs , frs∗ NCrs /X )m )
(B.13)
Now let us compute the cohomology groups. Recall that TCrs O(2) and NCrs /X O(−a) ⊕ O(a − 2). Without loss of generality, we can assume a ≥ 1. Then we have H 0 (rs , frs∗ TCrs ) H 0 (P1 , O(2n)), H 0 (, frs∗ NCrs /X ) H 0 (P1 , O(−an) ⊕ O((a − 2)n)).
(B.14)
Moreover, using Kodaira-Serre duality , H 1 (rs , frs∗ TCrs ) H 0 (rs , frs∗ (TC∗rs ) ⊗ ωrs )∗ H 0 (P1 , O(−2 − 2n))∗ = 0, H 1 (rs , frs∗ NCrs /X ) H 0 (rs , frs∗ (NC∗rs /X ) ⊗ ωrs )∗ H 0 (P1 , O(an − 2) ⊕ O((2 − a)n − 2))∗ .
(B.15)
We can write down explicit generators as follows: H 0 (rs , frs∗ TCrs ) :
∂x1 , t∂x1 , . . . , t 2n ∂x1 , 0, if a = 1 0 ∗ , (B.16) H (rs , f NCrs /X ) : ∂x3 , t∂x3 , . . . t (a−2)n ∂x3 if a ≥ 2 dx2 dt, tdx2 dt, . . . , t n−2 dx2 dt, if a = 1 1 ∗ ∗ , H (rs , f NCrs /X ) : dx dt, tdx3 dt, . . . , t n−2 dx3 dt 3 dx2 dt, tdx2 dt, . . . , t an−2 dx2 dt if a ≥ 2
where t is an affine coordinate of so that f : −→C is locally given by x1 = t n . In terms of representations of S 1 , we have k 0 ∗ n H (rs , f TCrs ) ⊕k=−n ρ1 , n 0, if a = 1 , (B.17) H 0 (rs , f ∗ NCrs /X ) n(a−2)
⊕k=0 ρ3 − nk ρ1 , if a ≥ 2
n−1
k ⊕k=1 ρ2 + nk ρ1 ⊕ ⊕n−1 if a = 1 k=1 ρ3 + n ρ1 , H 1 (rs , f ∗ NCrs /X ) . k ρ , if a ≥ 2 ⊕an−2 + ρ 2 1 k=1 n Now we can finish the computation of the edge factors (B.13). We will consider the cases a = 1 and a ≥ 2 separately, n−1 n−1 e(H 1 (rs , frs∗ TX )m ) −2 k=1 (nρ2 + kρ1 ) k=1 (nρ3 + kρ1 ) a=1: , =H e(H 0 (, frs∗ TX )m ) ρ1n (−ρ1 )n ((n − 1)!)2 an−2 e(H 1 (rs , frs∗ TX )m ) k=1 (nρ2 + kρ1 ) −2 . a≥2: = H n(a−2) n n e(H 0 (rs , frs∗ TX )m ) ρ1 (−ρ1 ) ((n − 1)!)2 n=0 (nρ3 − kρ1 ) (B.18)
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Before comparing (B.9) and (B.18) one has to remember that the weights ρi , i = 1, 2, 3 are supposed to satisfy the condition ρ1 +ρ2 +ρ3 = 0 in order to preserve the lagrangian cycles. Using this condition, we can rewrite the expressions in (B.18) as functions of ρ1 , ρ2 only a=1:
e(H 1 (rs , frs∗ TX )m ) e(H 0 (rs , frs∗ TX )m ) = (−1)
n−1
a≥2:
H
−2
n−1
k=1 (nρ2
e(H 1 (rs , frs∗ TX )m ) e(H 0 (rs , frs∗ TX )m ) = (−1)1+n(a−2) H −2
n−1
+ kρ1 ) n−1 k=1 (nρ2 + (n − k)ρ1 ) , n n ρ1 (−ρ1 ) ((n − 1)!)2
k=1 (nρ2
+ kρ1 ) n−1 k=1 (nρ2 + (na − k)ρ1 ) . n n ρ1 (−ρ1 ) ((n − 1)!)2 (B.19)
Therefore we can conclude that 1 m 1 m e(H 1 (rs , frs∗ TX )m ) 1+n(a−2) e(H (r , TXr ) ) e(H (s , TXs ) ) = (−1) e(H 0 (rs , frs∗ TX )m ) e(H 0 (r , TXr )m ) e(H 0 (s , TXs )m )
(B.20)
for all a. The last element we need is a similar formula for the contributions of the automorphism groups. One can easily check that e(Aut(rs )m ) = e(Aut(r )m )e(Aut(s )m ).
(B.21)
Acknowledgements. We are very grateful to Antonella Grassi for collaboration at an early stage of this project, especially for invaluable help with the graph combinatorics in Section Three. We would also like to thank Mina Aganagic, Sheldon Katz, Amir Kashani-Poor, Melissa Liu, Marcos Mari˜no and Cumrun Vafa for helpful discussions and Carel Faber for kindly sending us the Maple implementation of the algorithm [7]. The work of D.-E.D. has been partially supported by DOE grant DOE-DE-FG02-96ER40959 and by an Alfred P. Sloan foundation fellowship. The work of B.F. has been partially supported by DOE grant DOE-DE-FG02-96ER40959. We would also like to acknowledge the hospitality of KITP Santa Barbara where part of this work has been done.
References 1. Aganagic, M., Mari˜no, M., Vafa, C.: All Loop Topological String Amplitudes from Chern-Simons Theory. Commun. Math. Phys. 247, 467–512 (2004) 2. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The Topological Vertex. http://arxiv.org/list/ hep-th/0305132, 2003 3. Behrend, K., Fantechi, B.: The Intrinsic Normal Cone. Invent. Math. 128, 45 (1997) 4. Diaconescu, D.-E., Florea, B., Grassi, A.: Geometric Transitions and Open String Instantons. ATMP 6, 619 (2002) 5. Diaconescu, D.-E., Florea, B., Grassi, A.: Geometric Transitions, del Pezzo Surfaces and Open String Instantons. ATMP 6, 643 (2002) 6. Diaconescu, D.-E., Florea, B.: Large N Duality for Compact Calabi-Yau Threefolds. http://arxiv.org/list/hep-th/0302076, 2003 7. Faber, C.: Algorithms for Computing Intersection Numbers on Moduli Spaces of Curves, with an Application to the Class of the Locus of Jacobians. In : New Trends in Algebraic Geometry, Cambridge: Cambridge Univ. Press., 1999 8. Gopakumar, R., Vafa, C.: On the Gauge Theory/Geometry Correspondence. ATMP 3, 1415 (1999) 9. Graber, T., Pandharipande, R.: Localization of Virtual Classes. Invent. Math. 135, 487 (1999)
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10. Graber, T., Zaslow, E.: Open String Gromov-Witten Invariants: Calculations and a Mirror ‘Theorem’. http://arxiv.org/list/hep-th/0109075, 2001 11. Iqbal, A.: All Genus Topological String Amplitudes and 5-brane Webs as Feynman Diagrams. http://arxiv.org/list/hep-th/0207114, 2002 12. Iqbal, A., Kashani-Poor, A.-K.: SU (N) Geometries and Topological String Amplitudes. http:// arxiv.org/list/hep-th/0306032, 2003 13. Katz, S., Liu, C.-C. M.: Enumerative Geometry of Stable Maps with Lagrangian Boundary Conditions and Multiple Covers of the Disc. ATMP 5, 1 (2001) 14. Kontsevich, M.: Enumeration of Rational Curves via Torus Actions. In: The Moduli Space of Curves, Progr. Math. 129, Boston, MA: Birkh¨auser 1995, pp. 335–368 15. Labastida, J.M.F., Mari˜no, M., Vafa, C.: Knots, Links and Branes at Large N. JHEP 11, 007 (2000) 16. Li, J., Tian, G.: Virtual Moduli Cycles and Gromov-Witten Invariants of Algebraic Varieties. J. Amer. Math. Soc. 11, 119 (1998) 17. Li, J.: A Degeneration of Stable Morphisms and Relative Stable Morphisms. http://arxiv. org/list/math.AG/0009097, 2000 18. Li, J.: A Degeneration Formula of GW-invariants. http://arxiv.org/list/math.AG/0110113, 2001 19. Li, J., Song, Y.S.: Open String Instantons and Relative Stable Morphisms. ATMP 5, 67 (2002) 20. Liu, C.-C. M.: Moduli of J -Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an S 1 -Equivariant Pair. http://arxiv.org/list/math.SG/0210257, 2002 21. Liu, C.-C. M., Liu, K., Zhou, J.: On a Proof of a Conjecture of Mari˜no-Vafa on Hodge Integrals. Math. Res. Lett. 11(2), 259–272 (2004) 22. Liu, C.-C. M., Liu, K., Zhou, J.: A Proof of a Conjecture of Mari˜no-Vafa on Hodge Integrals. J. Diff. Geom. 65, 289–340 (2004) 23. Mari˜no, M., Vafa, C.: Framed Knots at Large N. http://arxiv.org/list/hep-th/0108064, 2001 24. Mayr, P.: Summing up Open String Instantons and N = 1 String Amplitudes. http:// arxiv.org/list/hep-th/0203237, 2001 25. Okounkov, A., Pandharipande, R.: Hodge Integrals and Invariants of the Unknot. Geom. Topol. 8, 675–699 (2004) 26. Ooguri, H., Vafa, C.: Knot Invariants and Topological Strings. Nucl. Phys. B 577, 419 (2000) Communicated by M.R. Douglas
Commun. Math. Phys. 257, 151–167 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1292-y
Communications in
Mathematical Physics
Orbifold Aspects of the Longo-Rehren Subfactors Nobuya Sato Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Tokyo 171-8501, Japan. E-mail: [email protected] Received: 7 May 2004 / Accepted: 19 August 2004 Published online: 12 April 2005 – © Springer-Verlag 2005
Abstract: In this article, we will prove that the subsectors of α-induced sectors for ˆ ⊃ M form a modular category, where M G ˆ is the crossed product of an infiM G ˆ nite factor M by the group dual G of a finite group G acting on M. In fact, we will prove that it is equivalent to M¨uger’s crossed product. By using this identification, we will exhibit an orbifold aspect of the quantum double of (not necessarily non-degenerate) obtained from a Longo-Rehren inclusion A ⊃ B under certain assumptions. We will apply the above description of the quantum double of to the ReshetikhinTuraev topological invariant of closed 3-manifolds, and we obtain a simpler formula, which is a generalization to the degenerate case of Turaev’s theorem that the Reshetikˆ is the product of hin-Turaev invariant for the quantum double of a modular category ˆ the Reshetikhin-Turaev invariant of and its complex conjugate. 1. Introduction Orbifold phenomena have repeatedly appeared in subfactor theory. The first appearance was to construct subfactors with Dynkin diagrams of type D2n out of subfactors with Dynkin diagrams of type A4n−3 through Dynkin diagram automorphisms [15]. This method essentially suggested that the orbifold construction removes the degeneracy of the braiding, and it is in fact proved in [10]. Along the same line, in [8] Evans and Kawahigashi extended the method of orbifold construction to the Hecke algebra subfactors of Wenzl [35]. In a more sophisticated way, Goto defined an orbifold subfactor as a simultaneous crossed product by non-strongly outer automorphism with trivial Loi invariant [11]. There is another known way to remove degeneracy of braiding, namely the quantum double construction. Originally, it is a way to construct a higher symmetric Hopf algebra out of the initial Hopf algebra and its dual Hopf algebra. In subfactor context,
Supported by the Grants-in-Aid for Scientific Research, JSPS.
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we have Ocneanu’s asymptotic inclusion M∞ ⊃ M ∨ M op constructed from a hyperfinite II1 subfactor N ⊂ M. The finite system of M∞ -M∞ bimodules obtained from an asymptotic inclusion is known to form a modular category. In fact, it is the so-called center construction in category theory [14]. However, this correspondence is not obvious because Ocneanu’s construction of the finite system of M∞ -M∞ bimodules makes an ingenious use of topological quantum field theory in three dimensions in the sense of Atiyah [1]. For infinite factors, Longo and Rehren introduced an interesting subfactor nowadays called the Longo-Rehren subfactor. The Longo-Rehren subfactor produces the same tensor category as the one the asymptotic inclusion does [22]. In his paper [13], Izumi examined and clarified the structure of a Longo-Rehren subfactor and its quantum double in a completely algebraic way, i.e., without using any help of TQFT. Moreover, he proved that the quantum double obtained from a LongoRehren inclusion is a modular category and further gave the description of modular Sand T -matrices in the language of sectors. Thus, the quantum double in subfactors also provides a machinery to remove degeneracy of braiding. See [10, 23] for the relationship between the orbifold construction and the asymptotic (or Longo-Rehren) inclusion. In [10], Evans and Kawahigashi proved that the quantum double of a finite system of ˆ is equivalent to ⊗ ˆ ˆ op as tensor categories. bimodules with non-degenerate braiding ˆ is It often happens that a subsystem of a finite system of non-degenerate braiding ˆ degenerate. A typical example is = a full system of WZW SU (N )k -model and = ˆ (The grading is introduced by the cyclic group ZN acting on the grading 0 part of . the set of integrable highest weight modules of level k [17].) In the case of SU (2)k and ˆ [10]. Later, SU (3)k , they succeeded to describe the quantum double of in terms of by using sector theory, Izumi obtained the quantum double of in the case of SU (N )k which description is quite close to M¨uger’s crossed product, namely dividing the douˆ ⊗ ˆ op by the group symmetry ZN . In this paper, we will generalize ble category Izumi’s argument to obtain the description of the quantum double of in the language ˆ is a minimal non-degenerate of M¨uger’s crossed product, under the assumption that ˆ ∩ = ∩ . extension i.e., M¨uger’s theory of crossed product has its origin at a conjecture by Rehren [30]: Extending endomorphisms on the observable algebra to the ones on the field algebra removes the degeneracy of the braiding. M¨uger solved this conjecture in [24] and he noticed that it could be possible to formulate the whole theory in terms of tensor category [25]. His formulation crucially depends on Doplicher-Roberts duality theory [6, 7]. (See [37] for another equivalent approach to crossed products.) It should be mentioned that at almost the same time as M¨uger’s work, Brugui`eres developed how to construct a modular category out of a certain ribbon category, based on Deligne’s internal characterization of the Tannakian category in characteristic 0. These modularizations have obvious applications to the Reshetikhin-Turaev TQFT. Brugui`eres himself examined some cases such as SL(N ) and P SL(N ) as examples [5]. Sawin used M¨uger’s machinery to obtain a modular category out of closed subsets of the Weyl alcove of a simple Lie algebra, which is essentially the case dividing some ribbon categories associated with simple Lie algebras by the cyclic group actions. He also obtained a topological invariant of closed 3-manifolds associated with such modular categories [33]. Since we have Longo-Rehren inclusions A ⊃ B ⊃ Bˆ for a minimal non-degenerˆ ⊃ , we can construct the Reshetikhin-Turaev invariant from the data ate extension of the quantum double of . As an application of an orbifold aspect of the inclusions
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A ⊃ B ⊃ Bˆ , we will have a simpler description of the Reshetikhin-Turaev invariant of closed 3-manifolds constructed from the quantum double of . This article is organized as follows: In Sect. 2, we collect some terminologies we need in this article. In particular, we will make quick (and somewhat brutal) reviews on the α-induction and M¨uger’s theory of crossed product. In Sect. 3, we explicitly compute ˆ ⊃ M, where M G ˆ is the crossed product the α-induction for the subfactor M G factor by the group dual. We will prove that the subsectors of α-induced sectors for ˆ ⊃ M form a modular category. This result is a folklore among the experts. We M G remark that there are some overlaps with [28]. In Sect. 4, we construct the Longo-Rehren ˆ ⊃ , and we inclusions A ⊃ B ⊃ Bˆ from a minimal non-degenerate extension ˆ will prove that B ⊃ Bˆ is conjugate to Bˆ G ⊃ Bˆ . This implies that the quantum ˆ and M¨uger’s crossed product. In Sect. 5, we will double of can be described by apply the result obtained in Sect. 4 to the Reshetikhin-Turaev invariant constructed from the quantum double of . Combined with the result in [16], we can have the statement that the Turaev-Viro-Ocneanu invariant constructed from is described by the sum of ˆ and its complex conjugate, the product of the framed link invariant constructed from which gives a special case of Ocneanu’s theorem [29]. 2. Preliminaries 2.1. Braided system of endomorphisms. Basics on sector theory and infinite subfactors [18, 19]. Let M, N be infinite factors, and we denote by Mor(N, M)0 the set of unital normal ∗-homomorphisms from N to M whose image has finite index. The statistical dimension d(ρ) of ρ ∈ Mor(N, M)0 is given by d(ρ) = [M : ρ(N )]1/2 . ρ ∈ Mor(N, M)0 is called irreducible if M ∩ ρ(N ) ∼ = C1M . For ρ, σ ∈ Mor(N, M)0 , the intertwiner space Hom(ρ, σ ) is defined by Hom(ρ, σ ) = {V ∈ M|Vρ(x) = σ (x)V , x ∈ N }. For every ρ ∈ Mor(N, M)0 , there are ρ¯ ∈ Hom(M, N )0 and isometries Rρ ∈ Hom(id, ρρ), ¯ 1 1 ¯ satisfying R¯ ρ∗ ρ(Rρ ) = d(ρ) , Rρ∗ ρ( ¯ R¯ ρ ) = d(ρ) . ρ¯ is called the conR¯ ρ ∈ Hom(id, ρ ρ) jugate of ρ. The unitary equivalence class of ρ ∈ Mor(N, M)0 is called a sector, and we denote by [ρ] the sector of ρ. Let M ⊃ N be an inclusion of infinite factors with finite index λ = [M : N ] and γ be its canonical endomorphism. Then, it is known [19] that there exist isometries v ∈ Hom(id, γ ) and Hom(γ , γ 2 ) satisfying v ∗ w = w∗ γ (v) = λ−1/2 1, w∗ γ (w) = ww ∗ , γ (w)w = w2 .
(2.1)
Moreover, M = N v and the conditional expectation E from M onto N is given by E(x) = w ∗ γ (x)w. Braided system of endomorphisms. Let M be an infinite factor, and 0 be a system of irreducible endomorphisms in End(M)0 = Mor(M, M)0 . More specifically, 0 is a set of irreducible normal ∗-endomorphisms of M closed under the following sector operations: (i) Different elements in 0 are inequivalent. (ii) idM ∈ 0 . 0 such ξ¯ ∈ that [ξ ] = [ξ¯ ]. (iii) For every ξ ∈ 0 there exists ζ ζ (iv) There exist non-negative integers Nξ η such that [ξ ][η] = ⊕ζ ∈0 Nξ η [ζ ].
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We denote by the subset of End(M)0 whose element are finite direct sums of the elements in 0 as sectors. A system of endomorphisms 0 is called braided if for any λ, µ ∈ 0 there exists a unitary intertwiner ε(λ, µ) ∈ Hom(λ · µ, µ · λ) with ε(id, µ) = ε(λ, id) = 1 satisfying the following (the Braiding-Fusion equations): For any λ, µ, ν ∈ 0 , t ∈ Hom(λ, µ · ν), σ (t)ε(λ, σ ) = ε(µ, σ )µ(ε(ν, σ ))t, tε(σ, λ) = µ(ε(σ, ν))ε(σ, µ)σ (t), σ (t)∗ ε(µ, σ )µ(ε(ν, σ )) = ε(λ, σ )t ∗ , t ∗ µ(ε(σ, ν))ε(σ, µ) = ε(σ, λ)ρ(t)∗ .
(2.2) (2.3) (2.4) (2.5)
We call above ε a braiding on 0 . For a given braiding ε(λ, µ) on 0 , unitary intertwiners ε(µ, λ)∗ also satisfies the above conditions of the braiding. We will use the notations ε + (λ, µ) = ε(λ, µ) and ε − (λ, µ) = ε(µ, λ)∗ to emphasize the difference. Degenerate sectors [30]. A sector ξ ∈ is said to be degenerate if ε + (ξ, η) = ε− (ξ, η) for every η ∈ 0 . is said to be non-degenerate if idM is the only degenerate sector. We denote the set of all degenerate sectors in by d and the set of all irreducible sectors in d by d0 . Note that d is a symmetric C ∗ -tensor subcategory of with direct sums, subobjects and conjugates. For ξ ∈ d0 , φξ (ε(ξ, ξ )) = λξ ∈ C, where φξ is the standard left inverse of ξ . The ω polar decomposition of λξ is given by d(ξξ ) . It is easy to show that ωξ = ±1 for ξ ∈ d (more generally, for an object in a symmetric C ∗ -tensor category). d is said to be even if ωξ = 1 for every irreducible ξ ∈ d . We assume d is even in the sequel. Then, by Doplicher-Roberts duality theory, there exists a finite group G up to isomorphism such ˆ where G ˆ is a category of finite dimensional unitary representations of G. that d ∼ = G, α-induction [2–4, 36]. Let M ⊃ N be an inclusion of infinite factors with finite index and γ be its canonical endomorphism. Let 0 ⊂ End(N )0 be a braided system of endomorphisms with a braiding ε. We define the α-induced endomorphism of λ ∈ 0 αλ ∈ End(M) by αλ = γ −1 · Ad(ε(λ, θ )) · λ · γ , where θ = γ |N . The systematic use of α-induction was first made by Xu [36], and further studied in a series of papers by B¨ockenhauer and Evans [2–4]. We list some properties of the α-induction [2, 36]: (i) d(αλ ) = d(λ), (ii) αλ · αµ = αλ·µ for any λ, µ ∈ 0 , (iii) αµ · αλ = Ad(ε(λ, µ)) · αλ · αµ for any λ, µ ∈ 0 , (iv) If [λ] = [λ1 ] ⊕ [λ2 ], λ, λ1 , λ2 ∈ , then [αλ ] = [αλ1 ] ⊕ [αλ2 ], and (v) [αλ¯ ] = [αλ ], λ ∈ 0 . The α-induction on 0 is extended to the one on preserving the above properties. 2.2. Premodular categories and M¨uger’s crossed product. To define M¨uger’s crossed product, we need some terminologies from category theory. See [21] for the basics on C ∗ -tensor category and [26] for the full description of crossed product.
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Assumption 2.1. We assume that C is a C ∗ -tensor category with conjugate, direct sums, subobjects, irreducible unit object ι and a unitary braiding ε. We use the following notations which are popular in the context of the algebraic quantum field theory: We use small Greek letters ρ, σ, etc. for objects of C, and the tensor product is denoted by ρσ instead of ρ ⊗ σ . For operations of arrows, we denote the composition of arrows S ∈ Hom(ρ, σ ), T ∈ Hom(σ, τ ) by T ◦ S ∈ Hom(ρ, τ ), the tensor product of S ∈ Hom(ρ1 , σ1 ), T ∈ Hom(ρ2 , σ2 ) by S × T ∈ Hom(ρ1 ρ2 , σ1 σ2 ). We denote by C0 the set of isomorphism classes of irreducible objects. We remark that under Assumption 2.1 C is a ribbon category and we denote a twist for each irreducible object ρ ∈ C by ωρ . Since we assume that C has a conjugate ρ¯ for each object ρ, there are Rρ ∈ Hom(ι, ρρ) ¯ and R¯ ρ ∈ Hom(ι, ρ ρ) ¯ satisfying R¯ ρ∗ × idρ ◦ idρ × Rρ = idρ , Rρ ∗ × idρ ◦ idρ¯ × R¯ ρ = idρ . Then, the dimension of an irreducible object ρ is defined by d(ρ) = Rρ ∗ ◦ Rρ , which takes its value in [1, ∞). (This definition of the dimension extends to reducible objects.) If the set C0 is finite, the category is called rational. Then, its dimension is defined by dim C = ξ ∈C0 d(ξ )2 . In subfactor context, this is called the global index. When C is rational, then we set the complex number S (ξ, η)idι = (Rξ ∗ × R¯ η∗ ) ◦ (idξ¯ × (ε(η, ξ ) ◦ ε(ξ, η)) × idη¯ ) ◦ (Rξ × R¯ η ) for ξ, η ∈ C0 . One can prove that S (ξ, η) does not depend on the choice of representatives of ξ and η. If S is invertible, C is called modular. When C is modular, the matrices 1 C 3 − 21 Diag(ωξ ) S = dim C S , T = |C | are unitaries and satisfy the relations [30, 34] where C =
ξ ∈C0
S 2 = (ST )3 = C, T C = CT , d(ξ )2 ω(ξ )−1 and C = δξ,η¯ .
Definition 2.2. If C satisfies Assumption 2.1 and is rational, we say C is C ∗ -premodular. For a C ∗ -premodular category C and its full subcategory S, we define C ∩ S , a full subcategory of C, by Obj C ∩ S = {ρ ∈ C|ε(σ, ρ) ◦ ε(ρ, σ ) = idρσ for all σ ∈ S}. We dim C remark that if C is modular we have dim C ∩ S = dim S by Theorem 3.2 [27]. Let C be a C ∗ -premodular category and we set DC = C ∩ C . DC is a symmetric tensor ∗-category with conjugates, thus in particular ωξ = ±1 for all irreducible ξ . We assume that DC is even, i.e., twist ωξ = 1 for each irreducible object ξ . Then, by Doplicher-Roberts duality theory [6, 7], there is a finite group such that DC is equivalent to U (G), where U (G) is a category of finite dimensional unitary representations of G. In the following, we use the symbol for the tensor product of U (G). ˆ be Let F be an invertible functor from DC to U (G) which gives the equivalence, G ˆ the set of all isomorphism classes of irreducible objects in DC , {γk |k ∈ G} be a section of objects in DC such that γ0 = ι and Hk = F (γk ). We choose an orthonormal basis Nm
m,α kl {Vk,l }α=1 of Hom(γm , γk γl ). Then, a category C 0 DC is defined in the following manner.
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• Obj C 0 DC = Obj Cwith the same tensor product as C. • HomC0 DC (ρ, σ ) = k∈Gˆ HomC (γk ρ, σ ) ⊗ Hk . ˆ S ⊗ ψk ∈ HomC D (ρ, σ ) and T ⊗ ψl ∈ HomC D (σ, τ ), where Let k, l ∈ G, 0 C 0 C T ∈ Hom(γl ρ, σ ), S ∈ Hom(γk σ, τ ) and ψk ∈ Hk , ψl ∈ Hl . We define the composition of arrows S ⊗ ψk ◦ T ⊗ ψl ∈ HomC0 DC (ρ, τ ) by m
S ⊗ ψk ◦ T ⊗ ψ l =
Nkl ˆ α=1 k∈G
m,α m,α ∗ S ◦ idγk × T ◦ Vk,l × idρ ⊗ F (Vk,l ) (ψk ψl )
(2.6) and extend this linearly. ˆ S ⊗ ψk ∈ HomC D (ρ1 , σ1 ) and T ⊗ ψl ∈ HomC D (ρ2 , σ2 ), Let k, l ∈ G, 0 C 0 C where S ∈ Hom(γk ρ1 , σ1 ), T ∈ Hom(γl ρ2 , σ2 ) and ψk ∈ Hk , ψl ∈ Hl . We define the tensor product of arrows S ⊗ ψk × T ⊗ ψl ∈ HomC0 DC (ρ1 ρ2 , σ1 σ2 ) by m
Nkl
S ⊗ ψk × T ⊗ ψl = ⊗
m,α S × T ◦ idγk × ε(γl , ρ1 ) × idρ2 ◦ Vk,l × idρ1 ρ2
ˆ α=1 k∈G m,α ∗ ) (ψk F (Vk,l
ψl )
(2.7)
and extend this linearly. Let S ⊗ ψ ∈ HomC0 DC (ρ, σ ), where S ∈ Hom(γk ρ, σ ) and ψk ∈ Hk . We define the ∗-operation of the arrows (S ⊗ ψ)∗ ∈ HomC0 DC (σ, ρ) by (S ⊗ ψ)∗ = Rk ∗ × idρ ◦ idγ¯k × S ∗ ⊗ ψk ·, F (R¯ k ),
(2.8)
where is a unit vector in the trivial representation H0 ∼ = C such that ψ = ψ = ψ for all ψ ∈ Obj U (G). It turns out that C 0 DC is a C ∗ -tensor category with conjugates and direct sums. Remark 2.3. For C, we have another braiding ε− (λ, µ) = ε(µ, λ)∗ . When we need to clarify which braiding we used, we will write C 0,+ DC and C 0,− DC depending on the choice of the braiding ε+ and ε − , respectively. C 0 DC is not closed under subobjects in general. However, we can enlarge C 0 DC to be closed under the subobjects. Such a procedure is called closure in Definition 3.11 in [25]. We denote the closure of C 0 DC by C DC and call it the crossed product of dim C C by DC . We remark that dim C DC = dim DC . It is important to mention that CDC is a modular category due to Theorem 4.4 in [25]. 3. Muger’s ¨ Crossed Product and α-Induction for Subfactors Let M, and d be as in Subsect. 2.1, and we assume that 0 is a finite set. We further assume that d is even and d ∼ = U (G), where G is a finite group. Then, by Doplicherˆ which contains M as a Roberts duality theory there exists a factor, denoted by M G, subfactor with index |G|. See [6] and [7] for the detailed accounts.
Orbifold Aspects of the Longo-Rehren Subfactors
157
ˆ is generated by M and isometries {ψ , i = 1, · · · , d(σ ), We may assume that MG i d σ ∈ 0 } satisfying : (σ )
(ι)
ψ (ι) := ψ1 = 1,
(3.9)
(σ ) ∗ (σ ) ψi ψj = δi,j δσ,σ , d(σ ) (σ ) (σ ) ∗ = 1, i=1 ψi ψi (σ ) (σ ) ψi x = σ (x)ψi , x ∈ M, d(τ ) (τ,k) (ρ) (σ ) (τ ) ψi ψj = τ ∈d k=1 V(ρ,i)(σ,j ) ψk , 0 (σ¯ ) (σ ) ∗ = Rσ∗ ψi , ψi d(σ1 ) d(σ2 ) (σ2 ) (σ1 ) (σ2 ) ∗ (σ1 ) ∗ ψj ψi = ε(σ1 , σ2 ), i=1 j =1 ψj ψi
(3.10) (3.11) (3.12) (3.13) (3.14) (3.15)
(τ,k)
where V(ρ,i)(σ,j ) ∈ Hom(τ, ρ · σ ) and Rσ ∈ Hom(ι, σ¯ · σ ). (σ )
Remark 3.1. (1) It is known that {ψi , i = 1, . . . , d(σ ), σ ∈ d0 } is a left M-module basis. (σ ) (σ ) ˆ the conditional expectation E : M G ˆ −→ M (2) When x = σ,i ti ψi ∈ M G, (ρ) ∗
(σ )
is given by E(x) = t (ι) . By computations, one has E(ψi ψj ˆ : M]. λ = [M G
1 ) = δσ,ρ δi,j d(σ ) , where
ˆ −→ M to M. Let θ be the restriction of the canonical endomorphism γ : M G Lemma 3.2. Let v =
(σ ) (σ ) σ,i ti ψi
∈ HomM Gˆ (id, γ ). Then, we have the relations (σ ) (σ ) λ (σ ) ∗ = ∈ HomM (σ, θ ) and ψi = d(σ v. Furthermore, ti , ) ti (σ ) ∗ (ρ) ) λ (σ ) (σ ) ∗ = 1. i = 1, . . . , d(σ ) satisfy ti tj = δσ,ρ δi,j d(σ σ,i d(σ ) ti ti λ and
(σ ) ti
(σ ) ∗ d(σ )E(vψi )
(ρ) ∗
Proof. Applying the conditional expectation E to the equation vψj (ρ) ∗
ψj
(σ ) (σ ) σ,i ti ψi
, we have (ρ) ∗
E(vψj
)=
(σ )
(σ )
(ρ) ∗
ti E(ψi ψj
(ρ)
) = tj
σ,i (σ )
Therefore, ti =
=
(σ ) ∗ xψi ,
(σ ) ∗
= d(σ )E(vψi
x ∈ M, we have (σ ) ∗
vψi
1 . d(ρ) (σ ) ∗
). Multiplying v from the left of the equality ψi (σ ) ∗
σ (x) = vxψi
(σ ) ∗
= γ (x)vψi
(σ ) ∗
= θ(x)vψi
.
Apply the conditional expectation E to the above equality, then we have (σ ) ∗
E(vψi (σ )
Hence, ti
(σ ) ∗
= d(σ )E(vψi
(σ ) ∗
)σ (x) = θ (x)E(vψi
) ∈ HomM (σ, θ ).
).
σ (x)
158
N. Sato (σ ) ∗ (ρ) tj
Let us compute ti
(σ ) ∗ (ρ) tj
ti
, (ρ) ∗
= d(σ )d(ρ)E(ψi v ∗ )E(vψj (σ )
) (ρ) ∗
= d(σ )d(ρ)w ∗ γ (ψi v ∗ )ww ∗ γ (vψj (σ )
)w (ρ) ∗
= d(σ )d(ρ)w∗ γ (ψi )γ (v ∗ )ww ∗ γ (v)γ (ψj (σ )
= =
)w
(σ ) (ρ) ∗ λ−1 d(σ )d(ρ)w ∗ γ (ψi ψj )w (σ ) (ρ) ∗ λ−1 d(σ )d(ρ)E(ψi ψj )
= δσ,ρ δi,j (σ ) ∗
Next, we compute ti
d(σ ) . λ
v, (σ ) ∗
ti
v = d(σ )E(ψi v ∗ )v = d(σ )w ∗ γ (ψiσ )γ (v ∗ )wv (σ )
= λ−1/2 d(σ )w ∗ γ (ψi )v (σ )
= λ−1/2 d(σ )w ∗ vψi d(σ ) (σ ) ψi . = λ
(σ )
(σ ) ∗
(σ )
λ Hence, ψi = d(σ v. ) ti Then, we have the following identity
v=
(σ )
(σ )
ti ψi
=
σ,i
σ,i
λ (σ ) (σ ) ∗ t t v. d(σ ) i i
Multiplying v ∗ from the right of the above equality and applying E, then we have σ,i
This completes the proof.
λ (σ ) (σ ) ∗ = 1. t t d(σ ) i i
Proposition 3.3. Equation (3.15) is equivalent to the identity ε(θ, θ )v 2 = v 2 . λ (σ ) is isometry in Hom(σ, θ ). Hence, ε(θ, θ ) is given by Proof. By Lemma 3.2, d(σ ) ti ε(θ, θ ) =
σ,σ i,j
=
σ,σ i,j
λ (σ ) t σ d(σ ) j
λ (σ ) λ (σ ) ∗ λ (σ ) ∗ ε(σ, σ )σ ti tj t d(σ ) d(σ ) d(σ ) i (σ )
λ2 d(σ )−1 d(σ )−1 tj
(σ ) ∗
σ (ti )ε(σ, σ )σ (tj (σ )
(σ ) ∗
)ti
.
Orbifold Aspects of the Longo-Rehren Subfactors
159
Then, we have ε(θ, θ )v 2 =
σ,σ ,τ i,j,k
=
=
σ,σ i,j
(σ )
(σ )
(σ )
(σ )
(σ ) ∗
σ (ti )ε(σ, σ )σ (tj
λd(σ )−1 tj
σ,σ ,τ i,j,k
(σ )
λd(σ )−1 tj
(τ )
(σ ) ∗ (τ ) (σ ) (τ ) tk )ψi ψk
σ (ti )ε(σ, σ )σ (tj (σ )
σ (ti )ε(σ, σ )ψi ψj
tj
(σ ) (τ )
)ψi tk ψk
(σ )
(σ )
(3.16)
.
(σ )
(σ )
Equation (3.15) is equivalent to ε(σ, σ )ψi ψj = ψj ψi , and with this, the formula (3.16) is equal to (σ ) (σ ) (σ ) (σ ) (σ ) (σ ) (σ ) (σ ) tj σ (ti )ψj ψi = t j ψ j ti ψi = v 2 . (σ )
σ,σ i,j
(σ )
σ,σ i,j
(σ ) ∗
On the contrary, assume that ε(θ, θ )v 2 = v 2 . Multiplying σ (ti )∗ tj (σ ) (σ ) (σ ) (σ ) left of ε(θ, θ )v 2 = σ,σ i,j tj σ (ti )ε(σ, σ )ψi ψj , we have (σ )
(σ ) ∗
σ (ti )∗ tj (σ )
ε(θ, θ )v 2 =
d(σ )d(σ ) (σ ) (σ ) ε(σ, σ )ψi ψj . 2 λ (σ ) ∗
On the other hand, multiplying σ (ti )∗ tj (σ )
(σ ) ∗ 2
σ (ti )∗ tj (σ )
(σ )
Thus, ε(σ, σ )ψi ψj (σ )
(σ )
= ψj
v =
(σ )
ψi .
from the
from the left of v 2 , we have
d(σ )d(σ ) (σ ) (σ ) ψ j ψi . λ2
Remark 3.4. The identity ε(θ, θ )v 2 = v 2 is called the chiral locality condition in [4]. Lemma 3.5. For λ ∈ , we have αλ± (ψi ) = ε± (λ, σ )∗ ψi , (σ )
(σ )
(3.17)
where σ ∈ d 0 , i = 1, . . . , d(σ ). In particular, αλ+ = αλ− for λ ∈ ∩ d = {ρξ ∈ |ε(ξ, σ )ε(σ, ξ ) = 1, ∀σ ∈ d0 }. ˆ ⊃ M and θ the restriction of γ Proof. Let γ be the canonical endomorphism of M G to M. (σ ) (σ ) (σ ) ∗ Applying γ to (3.12), we have γ (ψi )θ (x) = θ · σ (x)γ (ψi ). Thus, γ (ψi ) ∈ Hom(θ · σ, θ ). By the Braiding-Fusion equation (2.5), ε± (λ, θ )∗ θ (ε ± (λ, σ )∗ )γ (ψi ) = λ(γ (ψi ))ε ± (λ, θ )∗ . (σ )
(σ )
Applying γ −1 , ε ± (λ, σ )∗ ψi
(σ )
= γ −1 · Ad(ε ± (λ, θ ))λ · γ (ψi ) = αλ± (ψi ). (σ )
(σ )
The last claim is clear because ε+ (λ, σ ) = ε− (λ, σ ) for λ ∈ ∩ d .
160
N. Sato
We have the following description of a set of intertwiners Hom(αλ , αµ ) (cf. Proposition 3.6 [28]). Lemma 3.6. For λ, µ ∈ , Hom(αλ , αµ ) = {
) d(σ
(σ )
(σ )
(σ )
ti ψi ; ti
∈ Hom(σ · λ, µ), i = 1, . . . , d(σ ), σ ∈ d0 }.
σ ∈d0 i=1
d(σ ) (σ ) (σ ) Proof. Let t ∈ Hom(αλ , αµ ). We may write t = σ ∈d i=1 ti ψi . We remark 0 that this expression is unique. For x ∈ M, we have (σ ) (σ ) (σ ) (σ ) ti ψi λ(x) = µ(x)ti ψi . σ,i (σ )
Since ψi
σ,i
∈ Hom(id, σ ), the above equality is (σ ) (σ ) (σ ) (σ ) ti σ · λ(x)ψi = µ(x)ti ψi . σ,i
(σ )
σ,i (σ )
Thus, ti σ · λ(x) = µ(x)ti for any x ∈ M, i = 1, . . . , d(σ ) and σ ∈ d0 . For t above, let us show tαλ (ψ (σ ) ) = αµ (ψ (σ ) )t, where ψ (σ ) is an isometry in (σ ) {ψi , i = 1, . . . , d(σ ), σ ∈ d0 }. We will show that the left-hand side is equal to the right-hand side,
ti ψi ε(λ, σ )∗ ψ (σ ) = (σ )
(σ )
σ,i
ti σ (ε(λ, σ )∗ )ψi ψ (σ (σ )
(σ )
)
σ,i
=
ε(µ, σ )∗ σ (ti )ε(σ, σ )ψi ψ (σ (σ )
(σ )
)
σ,i
=
ε(µ, σ )∗ σ (ti )ψ (σ ) ψi (σ )
(σ )
σ,i
=
ε(µ, σ )∗ ψ (σ ) ti ψi , (σ )
(σ )
σ,i
where we used the Braiding-Fusion equation (2.2) for the second equality. This completes the proof. Remark 3.7. By the above lemma, for ρ ∈ d0 we have d(ρ) (ρ) (ρ) (ρ) (ρ) Hom(id, αρ ) = ti ψi ; ti ρ(x) = ρ(x)ti , ∀x ∈ M, i = 1, . . . , d(ρ) i=1
(ρ) d(ρ) ∼ = SpanC {ψi }i=1 ,
which is a Hilbert space with dimension d(ρ). Since d(αρ ) = d(ρ), we conclude that d(ρ) αρ ∼ = ⊕i=1 id. This can be read that α-induction trivializes degenerate sectors.
Orbifold Aspects of the Longo-Rehren Subfactors
161
Let λ ∈ and we use the notation αλ instead of αλ+ = αλ− . We denote by α the ˆ 0 consisting of subsectors of αλ , when λ varies in . subset of End(M G) Thanks to Proposition 3.3, we can make full use of the arguments in Subsect. 3.3 in [4]. For this, let β, δ be subsectors of αλ and αµ for some λ, µ ∈ , respectively. We set εr (β, δ) = s ∗ αµ (t ∗ )ε(λ, µ)αλ (s)t ∈ Hom(β · δ, δ · β) with isometries t ∈ Hom(β, αλ ), s ∈ Hom(δ, αµ ). It is proved in Lemma 3.11 [4] that εr (β, δ) does not depend on λ, µ and on the isometries s, t. Moreover, εr (β, δ) for β, δ ∈ α defines a braiding (called a relative braiding) on α (Corollary 3.13 [4]). Under these preliminaries, we have the following Proposition 3.8. α is a modular category. p
q
Proof. Let αλ = ⊕i=1 βi , λ ∈ , and δj ∈ α such that αµ = ⊕j =1 δj for some µ ∈ α . Assume εr (βi , δj )εr (δj , βi ) = 1 for all j = 1, . . . , q. Then, we have ε(λ, δ)ε(δ, λ) = 1 by Lemma 3.14 [4]. Hence, for ∀δ ∈ , we have ε(λ, δ)ε(δ, λ) = 1, which implies λ ∈ d . d(λ) Since αλ = ⊕i=1 id by Remark 3.7, we have βi = id for all i = 1, . . . , p, which proves that εr is a non-degenerate braiding on α . Thus, α is modular. So far, we have discussed the similarities to M¨uger’s theory of crossed product. In fact, we have the following ˆ ⊃ M, the image of by the α ± -induction is Proposition 3.9. For the inclusion M G d α given by 0,± . In particular, is naturally identified with d . d(σ ) (σ ) (σ ) Proof. For the composition of the intertwiners, let s = ∈ σ ∈d0 i=1 si ψi d(ρ) (ρ) (ρ) Hom(αλ , αµ ), t = ρ∈d i=1 ti ψi ∈ Hom(αµ , αν ). Then, ts ∈ Hom(αλ , αν ) 0 defines the composition of morphisms t and s. We use the notations ρ = γk , σ = γl , s (l) = s (γl ) , t (k) = t (γk ) , ψ (l) = ψ (γl ) and (k) ψ = ψ (γk ) , for simplicity. It is enough to check the condition for t = t (k) ψ (k) and s = s (l) ψ (l) because of linearity, t (k) ψ (k) s (l) ψ (l) = t (k) γk (s (l) )ψ (k) ψ (l) =
m) d(γ
m,α (m) t (k) γk (s (l) )Vk,l ψα ,
γm ∈d0 α=1
which is (2.6). For the tensor product of the intertwiners, let s ∈ Hom(αλ1 , αµ1 ), t ∈ Hom(αλ2 , αµ2 ). Then, sαλ1 (t) ∈ Hom(αλ1 · αλ2 , αµ1 · αµ2 ). We compute sαλ1 (t) in the case s = s (k) ψ (k) and t = t (l) ψ (l) , sαλ1 (t) = s (k) ψ (k) t (l) ε(λ1 , γl )∗ ψ(l) = s (k) γk (t (l) )γk (ε(γλ1 , γl )∗ )ψ (k) ψ (l) m) d(γ m,α (m) = s (k) γk (t (l) )γk (ε(λ1 , γl )∗ )Vk,l ψα , γm ∈d0 α=1
which is (2.7).
162
N. Sato
For the ∗-operation of the intertwiners, let t = for t = t (σ ) ψ (σ ) , ∗
∗
(σ ) (σ ) σ,i ti ψi . We check the condition ∗
∗
(t (σ ) ψ (σ ) )∗ = ψ (σ ) t (σ ) = Rσ∗ ψ (σ¯ ) t (σ ) = Rσ∗ σ¯ (t (σ ) )ψ (σ¯ ) , which is (2.8). Thus, the image of the α-induction is the crossed product 0 d in the sense of M¨uger. The last claim is immediate from the definitions of d and α . 4. Longo-Rehren Inclusions A ⊃ B∆ ⊃ B∆ ˆ ˆ ⊃ a non-degenerate Let be a subset of End(M)0 with a finite braided system 0 , extension. The following definition was first introduced by Ocneanu [29]. ˆ ⊃ is called minimal if ˆ ∩ = d . Definition 4.1. A non-degenerate extension ˆ = dim dim d if the extension is minimal. Remark that we have dim ˆ ⊃ in the sequel. We assume the minimality of the non-degenerate extension ζ
N
ζ
ξ,η Let {T (ξ,η )i }i=1 be an orthonormal basis of Hom(ζ, ξ · η), ξ, η, ζ ∈ 0 . Let M be the opposite algebra of M and j : M −→ M op the anti-linear isomorphism. We set A = M ⊗ M op , ξ op = j · ξ · j , and ξˆ = ξ ⊗ ξ op . For the isometries {Vξ }ξ ∈0 ⊂ A satisfying ξ ∈0 Vξ Vξ∗ = 1, we define γ (x) = Vξ ξˆ (x)Vξ∗ .
ξ ∈0
Let V ∈ Hom(id, γ ), W ∈ Hom(γ , γ 2 ) be isometries defined by V = VidM , W =
ξ,η,ζ ∈0
d(ξ )d(η) ζ Vξ ξˆ (Vη )Tξ,η Vζ∗ , d(ζ ) dim
ζ Nξ,η ζ ζ ζ where Tξ,η = i=1 T (ξ,η )i ⊗ j (T (ξ,η )i ). Then, one can construct a subfactor B of A such that γ : A −→ B is the canonical endomorphism of the inclusion A ⊃ B [20]. We call the inclusion A ⊃ B the Longo-Rehren inclusion. In a similar manner, we can construct the Longo-Rehren inclusion A ⊃ Bˆ . By their constructions, we have the inclusions A ⊃ B ⊃ Bˆ . We define D() to be the set of endomorphisms ρ ∈ End(B )0 such that [ι ][ρ] is a finite direct sum of sectors in the decompositions of {[ξ ⊗ id op ][ι ]}ξ ∈0 , where ι is the inclusion map ι : B → A. We call D() the quantum double of . (For a categorical interpretation of the quantum double, see [26].) In Corollary 7.2 [13], it is ˆ is equivalent to ˆ ⊗ ˆ op as modular categories. proved that D()
Proposition 4.2. We assume that d ∼ = U (G), where G is an abelian group. Then, there exists an outer action α of G on Bˆ and the subfactor B ⊃ Bˆ is isomorphic to Bˆ α G ⊃ Bˆ .
Orbifold Aspects of the Longo-Rehren Subfactors
163
Proof. Let ι1 : Bˆ → B be the inclusion map. Then, by Theorem 7.4 in [13], we have +− ]. By the minimality of the non-degenerate extension, this is [¯ι1 ι1 ] = ⊕ξ ∈0 [ρ ξ,ξ¯ ˆ as groups and d(ρ¯ +− ) = d(ρξ ) = 1 for each [¯ι1 ι1 ] = ⊕ d [ρ¯ +− ]. Since G ∼ =G ξ ∈0
ξ,ξ
ξ,ξ
+− is an automorphism labeled by G. Then, by Theorem 4.1 in [12], there ˆ 0 , ρ ξ ∈ ξ,ξ¯ exists an outer action α of G on Bˆ and the dual inclusion of B ⊃ Bˆ is B ⊃ B G . Hence, B ⊃ Bˆ is isomorphic to Bˆ α G ⊃ Bˆ .
Theorem 4.3. Let D() be the quantum double of . Then, under the assumptions in ˆ ⊗ ˆ op ∩ d ) d , where the embedding ιd : d → Proposition 4.2, D() = ( ˆ ⊗ ˆ op is given by ιd (σ ) = (σ, σ op ). ˆ = ˆ ⊗ ˆ op by thanks to Corollary 7.2 in [13]. Proof. First, we may assume that D() ˆ Then, we may apply Proposition By its construction, M α G can be viewed as M G. op d ˆ ˆ op ∩ d ) d in End(B )0 . ˆ ˆ 4.2 to ⊗ ∩ to get the crossed product ( ⊗ By Lemma 7.6 in [13], the image of the α-induction in Proposition 3.9 is in D(). ˆ ⊗ ˆ op ∩ d ) d is a full subcategory of D(). Thus, ( ˆ ⊗ ˆ op ∩ d ) d , We compute the dimension of (
ˆ ⊗ ˆ op ∩ d ˆ ⊗ ˆ op dim dim = d d dim (dim )2
2 ˆ dim = = (dim )2 dim d
ˆ ⊗ ˆ op ∩ d ) d = dim(
= (dim D())2 , ˆ ⊃ for the fourth equality. where we used the minimality of the extension op d ˆ ⊗ ˆ ⊗ ˆ ∩ ) d , and this implies D() = ( Thus, dim D() = dim( op d d ˆ ∩ ) .
5. Application to the Reshetikhin-Turaev Invariants for 3-Manifolds We apply Theorem 4.3 to the Reshetikhin-Turaev invariant of 3-manifolds constructed from the quantum double D() to get a simpler description of it in this case. Before we state the theorem, we collect some general results on a premodular category. Lemma 5.1. Let M be a premodular category, P a non-degenerate extension of M and D be degenerates of M, i.e., D = M ∩ M . Then, for η, ζ ∈ P0 we have ω∈M0
Nηωζ¯ d(ω) = d(ηζ¯ )χM (ηζ¯ ),
where χM (ξ ) = 1 if ξ ∈ M0 otherwise.
(5.18)
164
N. Sato
Proof. We compute ξ ∈D0 S (ξ, η)S (ξ, ζ¯ ) in different ways. On one hand, we have S (ξ, η)S (ξ, ζ¯ ) = d(ξ )Nηωζ¯ S (ξ, ω) ξ ∈D0
ξ ∈D0 ω∈P0
=
ω∈P0
Nηωζ¯
=
d(ξ )S (ξ, ω)
ξ ∈D0
ω∈(P ∩D )0
Nηωζ¯ d(ω) dim D,
where we used ξ ∈D0 d(ξ )S (ξ, ω) = d(ω) dim DχP ∩D (ω) in Lemma 2.13 in [27] for the third equality. On the other hand, S (ξ, η)S (ξ, η) ¯ = S (ξ, ηζ¯ )d(ξ ) ξ ∈D0
ξ ∈D0
= d(ηζ¯ ) dim DχP ∩D (ηζ¯ ).
Thus, ω∈(P ∩D )0 Nηωζ¯ d(ω) = d(ηζ¯ )χP ∩D (ηζ¯ ) with P ∩ D = M implies the claim. Let C be a premodular category. Let L be a framed link with n components in the 3-sphere. We denote the invariant of the colored framed link by FC (L, λ), where λ = (λ1 , . . . , λn ) ∈ C0n . Set {L}C =
n
d(λi )FC (L; λ).
λ∈C0n i=1
We may assume that a closed 3-manifold M is obtained from surgery along the framed link L in the 3-sphere S 3 . Namely, M = ∂WL , where WL is the 4-manifold obtained by gluing n 2-handles to the 4-ball B 4 along L ⊂ S 3 = ∂B 4 . We denote the signature of WL by σ (L). Let C be a modular category and we set C = ξ ∈C0 tξ−1 d(ξ )2 and DC = (dim C)1/2 . The Reshetikhin-Turaev invariant τC is defined by −σ (L)−n−1
τC (M) = (C )σ (L) DC
{L}C .
See [34] for the details of the definition. Lemma 5.2. Let C be a premodular category with C ∩ C = D and L be a framed link with n components. Then, we have {L}C = (dim D)n {L}CD . Proof. This is immediate from Remarques 2.1 1) and Proposition 3.7 1) in [5].
ˆ and associated with We now go back in the case of braided C ∗ -tensor categories subfactors. Recall that we have assumed the minimality of the non-degenerate extension ˆ ⊃ . For λ, µ ∈ , ˆ we put 1 [λ, µ] = Nλνµ¯ d(ν). ˆ dim ν∈0
Orbifold Aspects of the Longo-Rehren Subfactors
165
Theorem 5.3. Let M be a closed 3-manifold obtained from surgery along the framed link L with n components. Then, the Reshetikhin-Turaev invariant for D() is given by τD() (M) =
n 1 [λi , µi ] Fˆ (L; λ)Fˆ (L; µ). dim n ˆ i=1 λ,µ∈ 0
Proof. Since D() = DD() , we have τD() (M) =
1 (dim )n+1
n
d(ξ˜i )FD() (L; ξ˜ ).
ξ˜ ∈D()n0 i=1
Then, by Theorem 4.3 and Lemma 5.2, τD() (M) =
1 n+1 (dim ) (dim d )n
n
d(ζ˜i )FD() (L; ζ˜ ).
(5.19)
ˆ ˆ op ∩d )n i=1 ζ˜ ∈(⊗ 0
ˆ 0 such that ζ˜ = λ ⊗ We note that for ζ˜ ∈ ( ⊗ op ∩ d )0 there exist λ, µ ∈ op µ . With this and Lemma 5.1, we have d(ζ˜ )χ∩ ¯ = d(λ)d(µ)χ ¯ ∩ ¯ = d (λµ) d (λµ) ˆ ˆ ˆ [λ, µ] dim Hence, the right-hand side of (5.19) is n ˆ n (dim ) [λi , µi ] (dim )n+1 (dim d )n n ˆ i=1 λ,µ∈ 0
op
op
op
×F⊗ ˆ ˆ op (L; λ1 ⊗ µ1 , λ2 ⊗ µ2 , . . . , λn ⊗ µn ). op
op
op
Since we have the equality F⊗ ˆ ˆ op (L; λ1 ⊗ µ1 , λ2 ⊗ µ2 , . . . , λn ⊗ µn ) = Fˆ (L; λ1 , . . . , λn )Fˆ (L; µ1 , . . . , µn ) and the minimality of the non-degenerate extenˆ ⊃ , we have sion τD() (M) =
n 1 [λi , µi ] Fˆ (L; λ)Fˆ (L; µ). dim n ˆ i=1 λ,µ∈ 0
Remark 5.4. With Theorem 5.2 in [16], which claims that the Turaev-Viro-Ocneanu invariant for is equal to the Reshetikhin-Turaev invariant for D(), Theorem 5.3 proves a slightly different statement of Theorem 3.2 in [29] in the special case, although Ocneanu claims that it also holds true for G, a non-abelian group. Acknowledgement. The author would like to thank Professors Y. Kawahigashi and M. Izumi for valuable discussions in the early stage of this work. He also thanks Professor Yamagami and Dr. M¨uger for comments on the preliminary version of this manuscript. The author would like to thank the referee whose suggestions made this article more natural.
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References 1. Atiyah, M.F.: Topological quantum field theories. Publ. Math. I.H.E.S. 68, 175–186 (1989) 2. B¨ockenhauer, J., Evans, D.E.: Modular invariants, graphs, and α-induction for nets of subfactors. I. Commun. Math. Phys. 197, 361–386 (1998) 3. B¨ockenhauer, J., Evans, D.E.: Modular invariants, graphs, and α-induction for nets of subfactors. II. Commun. Math. Phys. 200, 57–103 (1999) 4. B¨ockenhauer, J., Evans, D.E.: Modular invariants, graphs, and α-induction for nets of subfactors. III. Commun. Math. Phys. 205, 183–228 (1999) 5. Brugui`eres, A.: Cat´egories pr´emodulaires, modularisations et invariants de vari´et´es de dimension 3. Math. Ann. 316, 215–236 (2000) 6. Doplicher, S., Roberts, J.E.: Endomorphisms of C ∗ -algebras, crossed products and duality for compact groups. Ann. Math. 130, 75–119 (1989) 7. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98, 157–218 (1989) 8. Evans, D.E., Kawahigashi, Y.: Orbifold subfactors from Hecke algebras. Commun. Math. Phys. 165, 445–484 (1994) 9. Evans, D.E., Kawahigashi, Y.: On Ocneanu’s theory of asymptotic inclusions for subfactors, topological quantum field theories and quantum doubles. Internat. J. Math. 6, 205–228 (1995) 10. Evans, D.E., Kawahigashi, Y.: Orbifold subfactors from Hecke algebras II. Commun. Math. Phys. 196 , 331–361 (1998) 11. Goto, S.: Orbifold construction for non-AFD subfactors. Internat. J. Math. 5, 725–746 (1994) 12. Izumi, M.: Applications of fusion rules to classification of subfactors. Publ. RIMS 27, 953–994 (1991) 13. Izumi, M.: The structures of sectors associated with the Longo-Rehren inclusions I. General theory. Commun. Math. Phys. 213, 127–179 (2000) 14. Kassel, C.: Quantum groups. Berlin-Heidelberg-New York: Springer-Verlag, 1995 15. Kawahigashi, Y.: On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors. J. Funct. Anal. 127, 63–107 (1995) 16. Kawahigashi, Y., Sato, N., Wakui, M.: (2+1)-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariants. http://arvix.org/abs/math.OA/0208238, 2002. 17. Kohno, T., Takata, T.: Symmetry of Witten’s 3-manifold invariants for sl(n, C). J. Knot Theory and its Ramif. 2, 149–169 (1993) 18. Longo, R.: Index of subfactors and statistics of quantum fields I, II. Commun. Math. Phys. 126 217–247, (1989); 130 , 285–309 (1990) 19. Longo, R.: Duality for Hopf algebras and subfactors. I. Commun. Math. Phys. 159, 133–150 (1994) 20. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995) 21. Longo, R., Roberts, J.E.: A theory of dimension. K-theory 11, 103–159 (1997) 22. Masuda, T.: An analogue of Longo’s canonical endomorphism for bimodule theory and its application to asymptotic inclusions. Internat. J. Math. 8, 249–265 (1997) 23. Masuda, T.: Extension of automorphisms of a subfactor to the symmetric enveloping algebra. Internat. J. Math. 12 , 637–659 (2001) 24. M¨uger, M.: On charged fields with group symmetry and degeneracies of Verlinde’s matrix S. Ann. Inst. Henri Poincar´e, 71, 359–394 (1999) 25. M¨uger, M.: Galois theory for braided tensor categories and the modular closure. Adv. in Math. 150, 151–201 (2000) 26. M¨uger, M.: From subfactors to categories and topology II. The quantum double of tensor categories and subfactors. J. Pure Appl. Alg. 180, 159–219 (2003) 27. M¨uger, M.: On the structure of modular categories. Proc. London Math. Soc. 87, 291–308 (2003) 28. M¨uger, M.: Conformal orbifold models and braieded crossed G-categories. Commun. Math. Phys., DOI 10.1007/s00220-005-1291-z, 2005 29. Ocneanu, A.: Chirality for operator algebras. In: “Subfactors”, H. Araki, et al., (ed.), Singapore: World Scientific, 1994, pp. 39–63 30. Rehren, K.-H.: Braid group statistics and their super selection rules. In: “ The Algebraic Theory of Superselection sectors”, D. Kastler (ed.), Singapore: World Scientific, 1990 31. Rehren, K.-H.: Markov traces as characters for local algebras. Nucl. Phys. B(Proc. Suppl.) 18B, 259–268 (1990) 32. Roberts, J.E.: Crossed products of von Neumann algebras by group duals. Sympos. Math. XX, 335–363 (1976) 33. Sawin, S.F.: Jones-Witten invariants for nonsimply connected Lie groups and the geometry of the Weyl alcove. Adv. Math. 165, 1–34 (2002)
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34. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter, 1994 35. Wenzl, H.: Hecke algebras of type An and subfactors. Invent. Math. 92, 345–383 (1988) 36. Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 349– 403 (1998) 37. Yamagami, S.: Group symmetry in tensor categories and duality for orbifolds. J. Pure Appl. Alg. 167, 83–128 (2002) Communicated by Y. Kawahigashi
Commun. Math. Phys. 257, 169–192 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1293-x
Communications in
Mathematical Physics
Ionization for Three Dimensional Time-Dependent Point Interactions Michele Correggi1 , Gianfausto Dell’Antonio2, , Rodolfo Figari3 , Andrea Mantile4 1 2 3
International School for Advanced Studies, Trieste, Italy. E-mail: [email protected] Centro Linceo Interdisciplinare, Roma, Italy. E-mail: [email protected] Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II” and Sezione INFN, Napoli, Italy. E-mail: [email protected] 4 Dipartimento di Matematica e Applicazioni, Universit`a di Napoli “Federico II”, Napoli, Italy. E-mail: [email protected] Received: 11 May 2004 / Accepted: 20 September 2004 Published online: 18 February 2005 – © Springer-Verlag 2005
Abstract: We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the “strength” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t → ∞, starting from a bound state at time t = 0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states. 1. Introduction We shall study the time evolution of a three dimensional system with time-dependent Hamiltonian given by H (t) = H0 + HI (t), where the “perturbation” HI (t) is a zero-range interaction with time-dependent (periodic) “strength”. In particular we are interested in proving complete ionization of the system as t → ∞, starting from an initial condition at t = 0 given by a bound state of the system. By complete ionization one can mean two different statements. The weaker one is that the survival probability of the bound state, i.e. the square modulus of the scalar product of the state at time t with the bound state, goes to zero as t → ∞. The stronger one is that every state in the Hilbert space of the system is a scattering state (see for example [11, 14]) of H (t), i.e. for every compact set S ⊂ R3 , 2 1 t lim dτ d 3 x τ ( x ) = 0, t→∞ t 0 S t denoting the time evolution of the state . The last statement is related to the absence of eigenvalues of the Floquet operator associated to H (t) (see [15, 13, 20]).
On leave from Dipartimento di Matematica, Universit`a di Roma, “La Sapienza”, Italy.
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The usual way to deal with problems of this kind is by means of time-dependent perturbation theory and Fermi’s golden rule, which gives for the survival probability the well known exponential decay for each order n in the perturbative expansion. On the other hand simple examples of regular perturbations show that the survival probability decays to zero as a power-law (i.e. the limits t → ∞ and n → ∞ can not be interchanged). When the perturbation is not small, it is in general very difficult to solve the problem and find the law of decay. Therefore it is interesting to find models in which a non-perturbative solution exists and study the survival probability. In this paper we study one such model, in which HI (t) is given by a three dimensional point interaction. We shall see that it is possible to prove asymptotic complete ionization and find a power law decay for the survival probability, under generic condition on the scattering length1 . The one-dimensional version of the same problem has been widely analyzed in [4– 7], where complete ionization is proved under a suitable and very weak condition on the Fourier coefficients of the strength of the interaction. We shall see that the same genericity condition is also sufficient in three dimensions to have complete ionization of the system. As it will be clear from their definition, three-dimensional point interaction Hamiltonians are significantly different from the one-dimensional delta interaction Hamiltonians used by the authors in [4–7] to model the periodically perturbed system undergoing the ionization process. The relevant parameter is the “charge” q(t) (see (2.1)), i.e. the coefficient of the singular part of the solution, and not the survival probability as in [4]. Nevertheless the equation satisfied by the charge bears strong similarity with the one considered by the above mentioned authors. This circumstance allowed us to follow, sometimes very closely, their strategy. From a physical point of view, the model we are going to study is related to the strong laser ionization of Rydberg atoms2 , showing many features of experimental data. Indeed, despite the simplicity of the model, as in the one-dimensional case, it is possible to reproduce many effects of multiphoton ionization of excited hydrogen atoms by microwave field, with good agreement with experiments (see [8]). 2. The Model The model we are going to study is a quantum particle subjected to a time-dependent point interaction fixed at the origin in three dimensions, namely a system defined by the time-dependent self-adjoint Hamiltonian Hα(t) , D(Hα(t) ) = ∈ L2 (R3 ) ∃ qλ (t) ∈ C, ( x ) − qλ (t) G λ ( x ) ∈ H 2 (R3 ), √ λ λ − qλ (t) G x=0 = α(t) + q(t) , (2.1) 4π Hα(t) + λ = H0 + λ − q(t) G λ , (2.2) where λ ∈ R, λ > 0 and
√
e− λ|x −x | x − x ) = G ( 4π | x − x | λ
is the Green function of the free Hamiltonian H0 = −. 1 2
In three dimensions the parameter α(t) is proportional to the inverse of the scattering length. See the discussion contained in [4, 8] and references therein.
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The operator3 (2.2) has absolutely continuous spectrum if α(t) is positive, while, when α(t) < 0, there exists exactly one negative eigenvalue −(4π α(t))2 , with normalized eigenfunction √ 2|α(t)| e4πα(t)|x | x) ≡ . (2.3) ϕα(t) ( | x| It is well known (see [9, 10, 12, 18, 19]) that the operator (2.2) defines a time propagation U (t, s) given by a two-parameters unitary family, solving the time-dependent Schr¨odinger equation i
∂t = Hα(t) t ∂t
and
(2.4)
t
x ) = U (t, s) s ( x ) = U0 (t − s)s ( x) + i t (
dτ q(τ ) U0 (t − τ ; x),
(2.5)
s
where U0 (t) = exp(−iH0 t), U0 (t; x) is the kernel associated to the free propagator and the charge q(t) satisfies a Volterra integral equation for t ≥ s, √ t √ t U0 (τ )s (0) α(τ )q(τ ) q(t) + 4 πi dτ √ dτ = 4 πi . (2.6) √ t −τ t −τ s s We are interested in studying complete ionization of system defined by (2.2) and (2.4), starting from initial conditions x ) = ϕα(0) ( x) 0 (
(2.7)
x ) being the bound state4 of Hα(0) . ϕα(0) ( We shall assume that α(t) is a real periodic continuous function with period T . The meaningful parameter of the system is the negative lower bound of α(t). Indeed, if inf(α(t)) ≥ 0, the wave operator associated to (H0 , Hα(t) ) is unitary (see [19]) so that any initial state evolves into a scattering state (see also the remark at the end of Sect. 5). Hence we require that 1. α(0) < 0.
(2.8)
Continuity of α(t) guarantees that it can be decomposed in a Fourier series, for each t ∈ R+ , and the series converges uniformly on every compact subset of the real line. In terms of the Fourier coefficients of α(t), we assume 2π , αn e−inωt , {αn } ∈ 1 (Z), ω = 2. α(t) = T n∈Z
∗ . 3. αn = α−n
We start by noticing that from (2.6) we have t t U0 (τ )0 (0) √ √ |q(τ )| dτ √ dτ |q(t)| ≤ 4 π sup(|α|) +4 π √ t −τ t −τ 0 0 3 4
For a general review about point interactions see [1, 2] and references therein. In order to do this analysis we shall require that α(0) < 0.
(2.9)
(2.10)
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from which we deduce that η(t)−|q(t)| ≥ 0, if η(t) is the unique solution of the equation t t U0 (τ )0 (0) √ √ η(τ ) η(t) = 4 π sup(|α|) dτ √ dτ +4 π . (2.11) √ t −τ t −τ 0 0 Iterating (2.11) once and differentiating we obtain for η the differential equation 2 dη = 16π 2 sup(|α|) η + 16π 2 U0 (t)0 (0), dt
(2.12)
where the inhomogeneous term is finite at each time t with, at most, an integrable singularity at t = 0. We conclude that |q(t)| ≤ η(t) ≤ C e16π
2 (sup(|α|))2 t
.
(2.13)
As a consequence the Laplace transform of q(t), denoted by ∞ dt e−pt q(t) q(p) ˜ ≡ 0
2 exists analytic at least for (p) > 16π 2 sup(|α|) . Applying the Laplace transform to Eq. (2.6), one has
q(p) ˜ = −4π
i αk q(p ˜ + iωk) + f˜(p), p
(2.14)
k∈Z
where
√ 2 ∞ 2 e−ik t 2|α(0)| i −pt 3 ˜ f (p) ≡ dt e d k 2 π p 0 k + (4π α(0))2 R3
2|α(0)| ∞ k2 =8 dk 2 ip (k + (4πα(0))2 )(k 2 − ip) 0
√ 2|α(0)| 4πα(0) + −ip = 4πi −ip (4πα(0))2 + ip and with the choice of the branch cut for the square root along the negative real line: if p = eiϑ , √
p=
√ iϑ/2
e
(2.15)
with −π < ϑ ≤ π. By unitarity of the evolution (2.4), it follows that the Laplace transform of q(t) is indeed analytic on the open right half plane: Proposition 2.1. The Laplace transform of q(t), solution of (2.6), is analytic at least for
(p) > 0.
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Proof. Using the decomposition of the wave function at time t defined by (2.5), we can write the survival probability in the following way: θ (t) ≡ ϕα(0) , t 2 3 = ϕα(0) , e−iH0 t ϕα(0) L (R )
t
+i ϕα(0) ( x) ,
dτ q(τ ) U0 (t − τ ; x)
0
Let us define
Z1 (t) ≡ ϕα(0) , e
−iH0 t
L2 (R3 )
.
(2.16)
L2 (R3 )
ϕα(0)
. L2 (R3 )
By the usual dissipative estimate for the free propagator, one has Z1 (t) ≤ c1 t − 23 as t → ∞ for some constant c1 ∈ R. Hence Z1 (t) belongs to L1 (R+ ) and then its Laplace transform Z˜ 1 (p) is analytic at least for (p) > 0. The second piece of the scalar product is given by t x) , dτ q(τ ) U0 (t − τ ; x) Z(t) ≡ i ϕα(0) ( 0
t
=i
L2 (R3 )
dτ q(τ ) e−iH0 (t−τ ) ϕα(0) (0)
0
and taking the Laplace transform of Z(t), we have ˜ ˜ Z(p) = Z˜ 2 (p) q(p), where Z˜ 2 (p) ≡ −
√ 4 2π|α(0)| √ 4πα(0) − −ip
is analytic for (p) > 0 and never equal to 0, because of condition (2.8). Hence the Laplace transform of θ (t) is given by θ˜ (p) = Z˜ 1 (p) + Z˜ 2 (p) q(p). ˜ But θ (t) is a bounded function5 , because of unitarity of the evolution (2.4), and then its Laplace transform is analytic on the open right half plane. The claim then follows from analyticity of Z˜ 1 (p), Z˜ 2 (p) and absence of zeros of Z˜ 2 (p). A well known property of Volterra integral operators, with regular or weakly singular kernel, implies Proposition 2.2. The homogeneous equations associated to (2.6) has no non-zero solup tion in Lloc (R+ ), 1 ≤ p ≤ ∞. 5
Actually |θ (t)| ≤ 1, since the initial state is normalized.
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Proof. The proof (see e.g. [16]) exploits the fact that the n-fold iterated kernel is a contraction in any Lp (0, Tn ) with Tn increasing to infinity for increasing n. In the following sections we shall prove asymptotic complete ionization of the system under generic conditions on α(t). Although the result does not depend on the sign of the mean α0 of α(t), we have to discuss separately the case α0 < 0 and α0 ≥ 0, because of the slightly different form of Eq. (2.14). 3. Case I: α0 < 0 Since α(0) < 0, changing the energy scale, it is always possible to assume that α(t) satisfies the normalization 1 4. α(0) = αn = − (3.1) 4π n∈Z
Moreover we introduce another condition we shall use later on: let T be the right shift operator on 1 (N), i.e. (3.2) T a n ≡ an+1 , we say that α = {αn } ∈ 1 (Z) is generic with respect to T , if α˜ ≡ {αn }n>0 ∈ 1 (N) satisfies the following condition: ∞ e1 = 1, 0, 0, . . . ∈ T n α. ˜
(3.3)
n=0
For a detailed discussion of this genericity condition see [4]. If (3.1) holds, Eq. (2.14) becomes (at least for (p) > 0) √ √ 4π 2i 2π 1 − −ip q(p) ˜ =− (3.4) αk q(p ˜ + iωk) − √ √ 4π α0 + −ip 4π α0 + −ip 1 + ip k∈Z k=0
and by Proposition 2.1 its solution is analytic on the open right half plane. In the following section we shall extend Eq. (3.4) above to the imaginary axis and study the behavior of the solution there. 3.1. Behavior on the imaginary axis at p = 0. Setting qn (p) ≡ q(p ˜ + iωn), we obtain a sequence of functions on the strip I = {p ∈ C, 0 ≤ (p) < ω}. Setting q(p) ≡ {qn (p)}n∈Z , Eq. (3.4) can be rewritten q(p) = L(p) q(p) + g(p),
(3.5)
where
Lq n (p) ≡ −
4π αk qn+k (p), √ 4πα0 + ωn − ip k∈Z k=0
(3.6)
Ionization for Three Dimensional Time-Dependent Point Interactions
and g(p) = {gn (p)}n∈Z with gn (p) ≡ −
√ √ 2i 2π 1 − ωn − ip . √ 4πα0 + ωn − ip 1 + ip − ωn
175
(3.7)
From the explicit expression of the operator (3.6) and (3.7), it is clear that the coefficients of the equation fail to be analytic on the imaginary axis at p0 = ((4π α0 )2 − ωn0 )i, for some n0 ∈ Z and then the solution may be singular there. Since (p) ∈ [0, ω), one has (4πα0 )2 (4π α0 )2 − 1 < n0 ≤ (3.8) ω ω and then the singularity appears at most in the equation for qn0 (there is only one integer6 which satisfies the previous inequality) at p0 = ((4π α0 )2 − ωn0 )i. For instance, if ω > (4π α0 )2 , the pole may be at p0 = (4πα0 )2 i in the equation for q0 . Actually we have to distinguish the so called (see [4]) resonant case, i.e. when (4πα0 )2 = N ω for some N ∈ N, because in that case we can have a pole only at p = 0 and then the solution is immediately seen to be analytic on the whole imaginary axis except at most for p = 0. Let us first consider the behavior of the solution on the imaginary axis for p = 0, p0 . We are going to prove that the solution is in fact analytic there. We prove first an important property of the operator L: Proposition 3.1. For p ∈ I, (p) = 0, p = 0, p0 , L(p) is an analytic operator-valued function and L(p) is a compact operator on 2 (Z). Proof. Analyticity on the imaginary axis for p = 0, p0 easily follows from the explicit expression of the operator. Moreover L(p) can be written L(p) = b(p) αk T n+k , k∈Z k=0
where b(p) is the operator (b q)n (p) ≡ bn (p) qn (p) = −
4π qn (p) √ 4πα0 + ωn − ip
and T is the right shift operator on 2 (Z). Since T = 1, the series converges strongly to a bounded operator. Moreover b(p) is a compact operator on the imaginary axis for p = 0, p0 : b(p) is the norm limit of a sequence of finite rank operators, because limn→∞ bn (p) = 0. Hence the result follows for example from Theorem VI.12 and VI.13 of [17]. Proposition 3.2. There exists a unique solution qn (p) ∈ 2 (Z) of (3.5) and it is analytic on the imaginary axis for p = 0, p0 . Proof. The key point will be the application of the analytic Fredholm theorem to the operator L(p) (Theorem VI.14 of [17]), in order to prove that (I − L(p))−1 exists for p = 0, p0 . 6
In fact n0 must be non negative.
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Since there is no non-zero solution in L2loc (R+ ) of the homogeneous equation associated to (2.6) (see Proposition 2.2), then the homogeneous equation associated to (3.5) has only the trivial solution in 2 (Z). Moreover the operator L is compact and thus analytic Fredholm theorem applies. The result easily follows, because g(p) ∈ 2 (Z) and each gn (p) is analytic for p = 0, p0 . We can now study Eq. (3.5) in a neighborhood of p0 (if p0 = 0). An important preliminary result is the following Lemma 3.3. Let (2.9) and the genericity condition (3.3) be satisfied by {αn } and let {hn (p)} be a sequence such that hn (p) ≡
hn (p) √ 4πα0 + ωn − ip
belongs to 2 (Z \ {n0 }). The system of equations rn = −
4π √ 4πα0 + ωn − ip
αk−n rk + hn (p)
(3.9)
k∈Z k=n,n0
has a unique solution {rn } ∈ 2 (Z \ {n0 }) in a purely imaginary neighborhood of p0 , where n0 ∈ Z and p0 ∈ I, (p0 ) = 0, are defined in (3.8). Moreover, if hn (p) is analytic in a neighborhood of p0 , the solution is analytic in the same neighborhood. Proof. Equation (3.9) is of the form r = L r + h , where h ≡ {hn } belongs to 2 (Z \ {n0 }) and L is a compact operator (see Proposition 3.1). In order to apply analytic Fredholm theorem to the operator L , we need to prove that there is no non-zero solution in a neighborhood of p0 of the homogeneous equation. Suppose that the contrary is true, so that {Rn } ∈ 2 (Z \ {n0 }) is a non-zero solution of Rn = −
4π αk−n Rn . √ 4πα0 + ωn − ip k∈Z k=n,n0
Multiplying both sides of equation above by Rn∗ and summing over n ∈ Z \ {n0 }, one has 2 ωn − ip Rn = −4π Rn ∗ αk−n Rk n∈Z n=n0
n,k∈Z n,k=n0
and, since the right-hand side is real, 2 =0 ωn − ip Rn n∈Z n=n0
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for p = iλ, 0 < λ < ω, and then Rn = 0 for n < 0. Now suppose that R = 0 and let n1 ∈ N be such that Rn = 0, n < n1 , and Rn1 = 0 (hence n1 ≥ 0). Fixing Rn0 = 0, for each n < n1 the homogeneous equation gives ∞
αk−n Rk = 0
k=n1
or, setting k = n1 − 1 + k , for n ≥ 0, ∞
αk +n Rn1 −1+k = 0
k =1
which implies (see (2.9)), for each n ≥ 0, R , T nα
2 (N)
= 0,
where Rn = Rn∗1 −1+n and (· , ·) stands for the standard scalar product on 2 (N). If {αn } satisfies the genericity condition (3.3), R has to be orthogonal also to e1 and then Rn1 = 0, which is a contradiction. Therefore R = 0. The first part of the lemma then follows from analyticity of L (p) and analytic Fredholm theorem. Moreover if {hn (p)} is analytic in a neighborhood of p0 , analyticity of the solution is a straightforward consequence. Proposition 3.4. If {αn } satisfies (2.9) and the genericity condition with respect to T (3.3), the unique solution {qn } ∈ 2 (Z) of (3.5) is analytic on the imaginary axis except at most for p = 0. Proof. If (4π α0 )2 = N ω for some N ∈ N (resonant case) there is nothing to prove, since the coefficients of (3.5) fail to be analytic only at p = 0. On the other hand, in the non resonant case, Proposition 3.2 guarantees analyticity on the imaginary axis for p = 0, p0 . Therefore it is sufficient to study the behavior of the solution in a neighborhood of p0 , where the coefficients of (3.5) have a singularity. We are going to prove that in fact the solution is analytic at p0 . The strategy of the proof is to analyze separately the terms qn , n = n0 , n0 being defined in (3.8), and then prove that also qn0 is analytic in a neighborhood of p0 . By Lemma 3.3 there is a unique solution of the system tn = −
4π 4π αn0 −n αk−n tk − . √ √ 4πα0 + ωn − ip 4π α0 + ωn − ip k∈Z k=n,n0
Setting qn = rn + tn qn0 , n = n0 , on (3.5), one has 4π rn + tn qn0 = − αk−n rk + tk qn0 αn0 −n qn0 + √ 4πα0 + ωn − ip k∈Z k=n,n0
√ √ 1 − ωn − ip 2i 2π − , √ 4πα0 + ωn − ip 1 + ip − ωn
(3.10)
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and therefore the equation for {rn }, n = n0 becomes √ 4π i 1 − ωn − ip αk rn+k + √ rn = − √ 4π α0 + ωn − ip 2π 1 + ip − ωn k∈Z
(3.11)
k=0,−n
while qn0 satisfies the equation qn0
4π =− √ 4π α0 + ωn0 − ip
αk−n0 rk + tk qn0
k∈Z k=n0
√ i 1 − ωn0 − ip +√ 2π 1 + ip − ωn0
or αk−n0 tk qn0 = −4π αk−n0 rk − 4π α0 + ωn0 − ip + 4π k∈Z k=n0
k∈Z k=n0
√ 2i 2π . √ 1 + ωn0 − ip
Since the last term is analytic in a neighborhood of p0 and {tn }, {rn } ∈ 2 (Z \ {n0 }) are both analytic, as it follows applying Lemma 3.3 above to (3.10) and (3.11), it is sufficient to prove that αk−n0 , t˜k = 0, k∈Z k=n0
where
t˜n ≡ tn (p)p=p . 0
Assume that the contrary is true: from Eq. (3.10) we obtain 2 ∗ αn−n t˜n∗ αk−n t˜k − 4π t˜∗ 4π α0 + ωn − ip0 t˜n = −4π 0 n n∈Z n=n0
n,k∈Z n,k=n0 ,n=k
= −4π
n∈Z n=n0
t˜n∗ αk−n t˜k ,
n,k∈Z n,k=n0 ,n=k
where we have used condition 2 in (2.9). The previous equation implies (the right hand side is real) t˜n = 0, ∀n < N0 = ipω0 and then, since −1 < N0 < 0, t˜n = 0, ∀n < 0. Hence from (3.10) we have, ∀n < 0, αk−n t˜k + αn0 −n = 0. k≥0 k=n0
Now supposing without loss of generality that t˜0 = 0 and setting Tn = t˜n−1 , n = n0 + 1, and Tn0 +1 = 1, we obtain, ∀n ≥ 0, ∞ k=1
αk+n Tk = 0,
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and using the genericity condition (3.3) (as in the proof of Lemma 3.3) we get T1 = t0 = 0, which is a contradiction. In conclusion qn0 is analytic in a neighborhood of p0 : analyticity of qn , n = n0 is then a straightforward consequence of analyticity of {rn }, {tn } and decomposition qn = rn + tn qn0 . The proof is then completed, since rn and tn belong to 2 (Z \ {n0 }) in a neighborhood of p = p0 . 3.2. Behavior at p = 0. We shall now study the behavior of the solution of (3.5) on the imaginary axis at the origin. With the choice (2.15) for the branch cut of the square root, it is clear that we must expect branch points of q(p), ˜ solution of (3.4), at p = iωn, n ∈ Z, which should imply a branch point at p = 0 for each qn in (3.5). We are going to show that qn , n ∈ Z has a branch point at p = 0. The non-resonant case and the resonant one will be treated separately. Proposition 3.5 (non-resonant case). If (4πα0 )2 = N ω, ∀N ∈ N and {αn } satisfies (2.9) and (3.3)√(genericity condition), the solution of Eq. (3.5) has the form qn (p) = cn (p) + dn (p) p, n ∈ Z, in an imaginary neighborhood of p = 0, where the functions cn (p) and dn (p) are analytic at p = 0. Proof. Setting qn = rn + tn q0 , n = 0 and choosing a solution {tn } ∈ 2 (Z \ {0}) of the system of Eqs. (3.10) with n0 = 0, we obtain that {rn } must satisfy (3.11). It is easy to see that the result of Lemma 3.3 holds also in a neighborhood of p0 = 0 with n0 = 0, so that {rn }, {tn } ∈ 2 (Z \ {0}) are unique and analytic at p = 0. Thus it is sufficient to prove that q0 , which is a solution of √ √ 2i 2π (1 − −ip) 4π α0 + −ip + 4π αk tk q0 = −4π αk r k − 1 + ip k∈Z k=0
k∈Z k=0
has the required behavior near p = 0. First, setting tn0 = tn (p = 0), we have to prove that αk tk0 = −α0 , k∈Z k=0
but, assuming that the contrary is true and multiplying both sides of Eq. (3.10) by tn0 and summing over n ∈ Z, n = 0, one has √ ∗ ωn |tn0 |2 = −4π tn0 αk−n tk0 + 4π α0 n∈Z
∗
n,k∈Z n,k=0
and then, because of the genericity condition (3.3), tn0 = 0, ∀n ∈ Z \ {0}, which is impossible, since {tn } solves (3.10). Now, calling F ≡ 4π α k tk k∈Z k=0
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and G ≡ −4π
αk rk ,
k∈Z k=0
we have
√ √ 2i 2π (1 − −ip) 4π α0 + −ip + F q0 = G + 1 + ip
and q0 = F +
√
p G ,
where F is analytic in a neighborhood of p = 0, because of analyticity of F and G, and √ √ 2i −2πi (4πα0 + F + 1) + −i (1 + ip) G . (3.12) G ≡− (1 + ip)[(4πα0 + F )2 + ip]
√ The resonant case, i.e. 4πα0 = − ωN for some N ∈ N, is not so different from the non-resonant one and we shall prove that the solution has the same behavior at the origin. The proof is slightly different because we need to show the absence of a pole at p = 0: from (3.5) one has √ 4π i 1 − ωN − ip αk qn+k (p) + √ qN (p) = √ √ 2π 1 + ip − ωN ωN − ωN − ip k∈Z k=0
and the coefficients have a singularity at p = 0. We are going to prove that in fact the solution has no pole at the origin: proceeding as in the proof of Proposition 3.4, let us begin with a preliminary result, which takes the place of Lemma 3.3: Lemma 3.6. Let (2.9) and the genericity condition (3.3) be satisfied by {αn } and let {hn (p)} be a sequence such that hn (p) hn (p) ≡ √ √ ωN − ωn − ip belongs to 2 (Z \ {N}). The system of equations rn = √
4π √ ωN − ωn − ip
αk rn+k + hn (p)
(3.13)
k∈Z k=0,−n
has a unique solution {rn } ∈ 2 (Z \ {N}) in a purely imaginary neighborhood of p = 0. Moreover, if hn (p) is analytic in a neighborhood of p = 0, the solution is analytic in the same neighborhood.
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Proof. We shall proceed as in the proof of Proposition 3.4, separating the contribution of rN , which may be singular: setting rn = un + vn rN , n = 0, N , on (3.13), one has 4π αk un+k + vn+k rN un + vn rN = √ αN−n rN + √ ωN − ωn − ip k∈Z k=0,−n,N−n
√ √ 2i 2π 1 − ωn − ip +√ , √ ωN − ωn − ip 1 + ip − ωn and requiring that {vn }, n = 0, N, solves 4π vn = √ √ ωN − ωn − ip k∈Z
αk vn+k + √
4π αN−n , √ ωN − ωn − ip
(3.14)
k=0,−n,N−n
the equation for {un }, n = 0, N , becomes 4π un = √ √ ωN − ωn − ip k∈Z
√ i 1 − ωn − ip αk un+k + √ . (3.15) 2π 1 + ip − ωn
k=0,−n,N−n
Moreover rN satisfies rN = √
4π √ ωN − ωN − ip
√ i 1 − ωn − ip α k uk + v k rN + √ 2π 1 + ip − ωn k∈Z
k=0,−N
or
√ ωN − ωN − ip − 4π αk−N vk rN
k∈Z k=0,N
√ i 1 − ωn − ip = 4π . αk−N uk + √ 2π 1 + ip − ωn k∈Z
k=0,N
Applying the discussion contained in the proof of Lemma 3.3, it is not difficult to see that the solutions of Eqs. (3.15) and (3.14) are analytic in a neighborhood of the origin and belong to 2 (Z \ {0, N}). Therefore it remains to prove that (setting vn0 = vn (p = 0)) αk−N vk0 = 0 k∈Z k=0,N
but the argument in the proof of Proposition 3.4 excludes this possibility, if {αn } satisfies the genericity condition. The proof is then completed, because analyticity of rN implies analyticity of all rn , n = 0, N . Proposition 3.7 (Resonant case). If (4πα0 )2 = N ω, for some N ∈ N and {αn } satisfies (2.9) and (3.3)√(genericity condition), the solution of Eq. (3.5) has the form qn (p) = cn (p) + dn (p) p, n ∈ Z, in an imaginary neighborhood of p = 0, where the functions cn (p) and dn (p) are analytic at p = 0. Proof. See the proof of Proposition 3.5 and Lemma 3.6 above.
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3.3. Complete ionization in the generic case. Summing up the results about the behavior of the Laplace transform q(p) ˜ of q(t) we can state the following Theorem 3.8. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , as t → ∞, 3
|q(t)| ≤ A t − 2 + R(t),
(3.16)
where A > 0 and R(t) has an exponential decay, R(t) ∼
Ce−Bt
for some B > 0.
Proof. Propositions 3.2, 3.4 and 3.5 guarantee that q(p) ˜ is analytic on the closed right half plane, except branch point singularities on the imaginary axis at p = iωn, n ∈ Z. Therefore we can choose an integration path for the inverse of Laplace transform of q(q) ˜ along the imaginary axis like in [4]. Proposition 3.5 implies that the contribution of the branch point at p = 0 is given by the integral ∞ √ 2i dp p G (−p) e−pt , 0
where G , defined in (3.12), is a bounded analytic function on the negative real line: from the explicit expression of F and G and Eqs. (3.11) and (3.10), it is clear that G is analytic and limp→∞ G (−p) = 0 on the real line. So that the corresponding asymptotic behavior as t → ∞ is ∞ ∞ 3 √ √ −pt dp p G (−p) e ≤ C dp p e−pt = A t − 2 . 0
0
Let us consider now the contribution of branch points at p = iωn, n = 0: from Propositions 3.5 and 3.7 it follows that, in a neighborhood of p = 0, √ qn (p) = cn (p) + dn (p) p, where cn (p) and dn (p) are analytic at p = 0. Moreover using the decomposition qn = rn + tn q0 , n = 0, as in the proof of Proposition 3.5 and 3.7, and studying Eq. (3.10) for tn , we immediately obtain {dn } ∈ 1 (Z \ {0}), because of condition 2 in (2.9). Since qn (p) = q(p ˜ + iωn), the contribution of singularities at p = iωn, n = 0, is then given by iωn 2 dp dn (p − iωn) p − iωn ept n∈Z iωn−∞ n=0 ∞
= 2i
dp
0
and the series
dn (−p) eiωnt
√
p e−pt
n∈Z n=0
dn (−p) eiωnt
n∈Z n=0
converges uniformly to a bounded function of t, because {dn } ∈ 1 (Z \ {0}). Adding up the contributions of every branch cut, one obtains the required leading term in the asymptotic behavior. Indeed the rest function R(t) is given by the contribution of poles outside the imaginary axis and then shows an exponential decay as t → ∞.
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A straightforward consequence of Theorem 3.8 is that the scalar product (and thus the survival probability of the bound state) θ (t) = ϕα(0) , t 2 3 L (R )
tends to 0 when t → ∞: Corollary 3.9. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , the system shows asymptotic complete ionization and, as t → ∞, 3
|θ (t)| ≤ D t − 2 + E(t), where D > 0 and E(t) has an exponential decay. Proof. The Laplace transform of θ (t) can be expressed in the following way (see the proof of Proposition 2.1): ˜ θ˜ (p) = Z˜ 1 (p) + Z˜ 2 (p) q(p), ˜ where Z˜ 1 (p) is analytic on the closed √ right half plane and Z2 (p) has only a branch point at the origin of the form a1 + a2 p. Hence θ˜ (p) has the same singularities as q(p) ˜ and then its asymptotic behavior coincides with that of q(t), i.e. 3
|θ (t)| ≤ D t − 2 + E(t) for some constant D ∈ R and for a bounded function E(t) with exponential decay.
In the following we shall prove a stronger result about complete ionization of the system, namely that every state ∈ L2 (R3 ) is a scattering state7 for the operator Hα(t) , i.e. for any finite R 2 1 t dτ F (| x | ≤ R)U (τ, 0) = 0, (3.17) lim t→∞ t 0 where F (S) is the multiplication operator by the characteristic function of the set S ⊂ R3 and U (t, s) the unitary two-parameters family associated to Hα(t) (see (2.4)). In order to prove (3.17), we first need to study the evolution of a generic initial datum in a suitable dense subset of L2 (R3 ) and then we shall extend the result to every state using the unitarity of the evolution defined by (2.4) (see e.g. [9]). Proposition 3.10. Let ∈ C0∞ (R3 \ {0}) a smooth function with compact support away from 0 and q(t) be the solution of Eq. (2.6) with initial condition 0 = . If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , as t → ∞, 3
|q(t)| ≤ A t − 2 + R(t),
(3.18)
where A > 0 and R(t) has an exponential decay, R(t) ∼ Ce−Bt for some B > 0. 7
For the definition of scattering states of a time-dependent operator see e.g. [11, 14].
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Proof. The proof of Proposition 2.1 still applies, considering θ (t) ≡ , t 2 3 L (R )
instead of θ (t), so that q(p), ˜ solution of (2.14) with initial condition 0 = , is analytic ∀p with (p) > 0. Hence we can consider the Laplace transform of Eq. (2.6), which has the form (2.14) with ∞ 2 i −pt e−ik 2 t , ˆ k) f (p) = dt e d 3 k ( 3 π p 0 R is the Fourier transform of ( ˆ k) where ( x ). The equation for q(p) ˜ is then given by 4π g(p) q(p) ˜ =− αk q(p ˜ + iωk) + , √ √ 4πα0 + −ip 4π α0 + −ip k∈Z k=0
where
g(p) =
2 π
∞
dt e 0
−pt
R3
e−ik 2 t . ˆ k) d 3 k (
It is now sufficient to show that the solution q(p) ˜ is also analytic on the imaginary axis except at most square root branch points at p = iωn as in the discussion of Sects. 3.2 and 3.3. is a smooth function with ˆ k) For every smooth function with compact support, ( an exponential decay as k → ∞, so that ˆ k) ˆ k) ( ( 2 2 3 g(is) = lim = −i d k d 3 k 2 + π R3 r + (s + k )i π R3 s + k2 r→0 is a bounded function for s > 0. Hence the function g(p) has no pole for (p) ∈ (0, ω) and therefore the result contained in Proposition 3.4 still holds. Moreover ∞ ˆ k) 2 2 ( 3 ˆ −ik 2 t g(0) = d k (k) dt e = −i d 3 k 2 π R3 π R3 k 0 which is again bounded, so √ that g(p) has at the origin at most a branch point singularity of the form a(p) + b(p) p: following the proofs of Proposition 3.5 and 3.7, we can show that q(p) ˜ has the same behavior at the origin. In conclusion the solution is analytic on the closed√right half plane except for branch points at p = iωn, n ∈ Z, of the form a(p) + b(p) p − iωn. The proof of Theorem 3.8 then implies that q(t) has the prescribed behavior as t → ∞. Theorem 3.11. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , every ∈ L2 (R3 ) is a scattering state of Hα(t) , i.e. for any finite R 2 1 t dτ F (| x | ≤ R)U (τ, 0) = 0. lim t→∞ t 0
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Proof. We shall restrict the proof to the dense subset of L2 (R3 ) given by smooth functions with compact support and then we shall extend the result to every state using the unitarity of the evolution defined by (2.5) (see e.g. [9]). Actually we are going to prove a stronger statement, i.e. ∀ε > 0, there exists t0 such that ∀t > t0 , F (| x | ≤ R)U (t, 0) ≤ ε. The evolution of an initial state according to (2.5) is given by t t ( x ) = U (t, s)s ( x ) = U0 (t − s)s ( x) + i dτ q(τ ) U0 (t − τ ; x). (3.19) s
Moreover, since t ∈ D(Hα(t) ), the following decomposition holds: x ) = ϕt ( x) + t (
q(t) , 4π | x|
(3.20)
2 (R3 ) and where q(t) is the solution of (2.6), ϕt ∈ Hloc
ϕt (0) = α(t)q(t). We are going to show that, if q(t) ∈ L1 (R+ ), t satisfies the required property. Let us start analyzing the second term in (3.19): imposing the unitarity condition of the evolution we have 2 t 2 2 x) + i dτ q(τ ) U0 (t − τ ; x) s = t = U0 (t − s)s ( s
and then 2 t t dτ q(τ ) U (t − τ ; x ) = 2 dτ q(τ ) U (t − τ ; x ) , U (t − s) ( x ) 0 0 0 s s s t = 2 dτ q ∗ (τ ) e−iH0 (τ −s) s (0) s
but, using the decomposition (3.20), e−iH0 (s−τ ) s (0) = e−iH0 (s−τ ) ϕs (0) +
= e−iH0 (s−τ ) ϕs (0) +
R3
d 3 k e−ik
2 (τ −s)
q(s) (2π )3 k 2
q(s) . √ 4π π i τ − s √
2 (R3 ), the absolute value of the first term on the right hand side is bounded Since ϕs ∈ Hloc by a constant c(τ, s) < ∞ such that c(s, s) = q(s) and
lim c(τ, s) = 0.
τ →∞
Hence there exists s1 (ε) > 0 such that, ∀s > s1 , t 2ε2 2 dτ q ∗ (τ ) e−iH0 (s−τ ) ϕs (0) ≤ 9 s
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if q(t) ∈ L1 (R+ ). Moreover by the same reason there exists s2 (ε) > 0 such that ∀s > s2 , t 2ε2 q(s) ∗ ≤ dτ q (τ ) √ √ . 2 9 4π πi τ − s s Setting s0 (ε) = max(s1 (ε), s2 (ε)), one has ∀s > s0 , t 2ε ≤ dτ q(τ ) U (t − τ ; x ) , 0 3 s
(3.21)
so that the whole L2 −norm of the second term in decomposition (3.19) is suitably small for s > s0 . On the other hand the first term in (3.19) is the free evolution of a L2 −function and hence there exists δ(ε) > 0 such that ∀t > s + δ and ∀R < ∞, ε F (| x | ≤ R)U (t − s)s ≤ . (3.22) 3 Setting t0 (ε) = s0 (ε) + δ(ε), from (3.19), (3.21) and (3.22) one has F (| x | ≤ R)t ≤ ε ∀t > t0 , if q(t) ∈ L1 (R+ ). By Proposition 3.10 the inequality is then satisfied by every ∈ C0∞ (R3 \ {0}): unitarity of the family U (t, s) allows to extend the result to the whole Hilbert space L2 (R3 ). Corollary 3.12. If {αn } satisfies (2.9) and the genericity condition with respect to T (3.3), the discrete spectrum of the Floquet operator associated to Hα(t) , K ≡ −i
∂ + Hα(t) ∂t
is empty. Proof. The result is a straightforward consequence of Theorem 3.11: every eigenvector of K differs from a periodic function by a phase factor and hence can not satisfy (3.17). 4. Case II: α0 = 0 If α(t) = α0 = 0 does not depend on time, the problem has a simple solution: the spectrum of Hα(t) is absolutely continuous and equal to the positive real line, with a resonance at the origin; hence there is no bound state and the system shows complete ionization irrespective of the initial datum. On the other hand if α(t) is a zero mean function, we shall see that the genericity condition (3.3) in our approach is still needed to prove complete ionization. So let us assume that α0 = 0, the normalization (3.1) holds and the initial datum is given by (2.7): Eq. (2.14) then becomes
√ i 2π i 1 − −ip (4.1) αk q(p ˜ + iωk) − 2i q(p) ˜ = −4π p p 1 + ip k∈Z k=0
√ with the choice (2.15) for the branch cut of p. By Proposition 2.1 the solution is analytic in the open right half plane. In the following section we shall study the singularities on the imaginary axis.
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4.1. Singularities on the imaginary axis. Setting qn (p) ≡ q(p ˜ + iωn), p ∈ I = [0, ω), as in Sect. 3.1, Eq. (4.1) assumes the form (3.5), q(p) = M(p) q(p) + o(p)
(4.2)
4π αk qn+k (p) Mq n (p) ≡ − √ ωn − ip
(4.3)
with
k∈Z k=0
and o(p) = {on (p)}n∈Z ,
√ 2i 2π on (p) ≡ − √ . √ ωn − ip (1 + ωn − ip)
(4.4)
Proposition 4.1. For p ∈ I, (p) = 0, p = 0, M(p) is an analytic operator-valued function and M(p) is a compact operator on 2 (Z). Proof. See the proof of Proposition 3.1.
Proposition 4.2. There exists a unique solution qn (p) ∈ 2 (Z) of (4.2) and it is analytic on the imaginary axis for p = 0. Proof. See the proof of Proposition 3.2.
Proposition 4.3. If {αn } satisfies (2.9) and the genericity condition (3.3), the solution of √ Eq. (4.2) has the form qn (p) = cn (p) + dn (p) p, n ∈ Z, in a neighborhood of p = 0, where the functions cn (p) and dn (p) are analytic at p = 0. Proof. Let us proceed as in the proof of Proposition 3.5: setting qn = rn + tn q0 , n ∈ Z \ {0}, where {tn } is the solution of 4π 4π α−n αk tn+k − √ tn = − √ . (4.5) ωn − ip ωn − ip k∈Z k=0,−n
A slightly different version of Lemma 3.3 guarantees that the solution {tn } ∈ 2 (Z \ {0}) is unique and analytic at p = 0. By means of this substitution we obtain √ 4π 2i 2π rn = − √ αk rn+k − √ (4.6) √ ωn − ip ωn − ip (1 + ωn − ip) k∈Z k=0,−n
and
√ 2i 2π 4π q0 = − √ α k rk + t k q0 − √ √ −ip −ip (1 + −ip) k∈Z k=0
or
√ 2 2π −ip + F q0 = G − , √ 1 + −ip
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where (like in the proof of Proposition 3.5) F ≡ 4π αk tk k∈Z k=0
and G ≡ −4π
α k rk .
k∈Z k=0
Moreover F (0) = 0, because of genericity condition (3.3) (see the proof of Proposition 3.5), F and G are analytic in a neighborhood of p = 0 (see Lemma 3.3), so that √ q0 = F + p G , where F and G are analytic and √ √ 2 −2πi(F + 1) − −i(1 + ip)G . G ≡ (1 + ip)(F 2 + ip) 4.2. Complete ionization in the generic case. As in Sect. 3 we can now state the main result: Theorem 4.4. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , as t → ∞, 3
|q(t)| ≤ A t − 2 + R(t),
(4.7)
where A > 0 and R(t) has an exponential decay, R(t) ∼ Ce−Bt for some B > 0. Proof. See the proof of Theorem 3.8.
Corollary 4.5. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , the system shows asymptotic complete ionization and, as t → ∞, 3
|θ (t)| ≤ D t − 2 + E(t), where D > 0 and E(t) has an exponential decay. Proof. See the proof of Corollary 3.9.
Theorem 4.6. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , every ∈ L2 (R3 ) is a scattering state of Hα(t) , i.e. for any finite R 2 1 t dτ F (| x | ≤ R)U (τ, 0) = 0. lim t→∞ t 0 Moreover the discrete spectrum of the Floquet operator is empty. Proof. See the proof of Proposition 3.10 and Theorem 3.11.
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5. Case III: α0 > 0. To complete the analysis of the problem, we shall consider the case of mean greater than 0: taking the normalization (3.1) and the initial condition (2.7), (2.14) assumes the form (3.4): √ √ 4π 2i 2π 1 − −ip q(p) ˜ =− αk q(p ˜ + iωk) − . (5.1) √ √ 4π α0 + −ip 4π α0 + −ip 1 + ip k∈Z k=0
Analyticity of the solution on the open right half plane is a consequence of Proposition 2.1. Moreover, following the discussion contained in Sect. 3 and setting qn (p) ≡ q(p ˜ + iωn), (p) ∈ [0, ω), the equation assumes the form (3.5). Let us now consider the behavior √ on the imaginary axis: singularities for (p) = 0 are associated to zeros of 4πα0 + ωn + s, s ∈ [0, ω), but, since α0 > 0, it is clear that the expression can not have zeros on the imaginary axis. Hence the proof of Proposition 3.2 can be extended to the closed right half plane except the origin: Proposition 5.1. If {αn } satisfies (2.9), the solution q(p) ˜ of (5.1) is unique and analytic for (p) ≥ 0, p = iωn, n ∈ Z. Proof. See the proof of Proposition 3.2, Propositions 3.1 and 2.2 and the previous discussion. Moreover the behavior at the origin is described by the following Proposition 5.2. If {αn } satisfies (2.9) and the genericity condition with respect to T (3.3), then, in an imaginary neighborhood √ of p = iωn, n ∈ Z, the solution of Eq. (5.1) has the form q(p) ˜ = cn (p) + dn (p) p − iωn, where the functions cn (p) and dn (p) are analytic at p = iωn. Proof. The proof of Proposition 3.5 still applies with only one difference: since, independently of ω, the solution can not have a pole on the imaginary axis, we need not to distinguish between the resonant case and the non-resonant one. We can now prove asymptotic complete ionization of the system: Theorem 5.3. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , as t → ∞, 3
|q(t)| ≤ A t − 2 + R(t),
(5.2)
where A > 0 and R(t) has an exponential decay, R(t) ∼ Ce−Bt for some B > 0. Moreover the system shows asymptotic complete ionization and, as t → ∞, 3
|θ (t)| ≤ D t − 2 + E(t), where D > 0 and E(t) has an exponential decay. Proof. See the proof of Theorem 3.8 and Corollary 3.9.
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Theorem 5.4. If {αn } satisfies (2.9) and the genericity condition (3.3) with respect to T , every ∈ L2 (R3 ) is a scattering state of Hα(t) , i.e. for any finite R 1 t→∞ t
t
lim
2 dτ F (| x | ≤ R)U (τ, 0) = 0.
0
Moreover the discrete spectrum of the Floquet operator is empty. Proof. See the proof of Proposition 3.10 and Theorem 3.11.
Remark 5.5. If α(t) ≥ 0, ∀t ∈ R+ , Proposition 5.2 holds without the genericity condition on the Fourier coefficients of α(t): for instance the genericity condition enters (see the proof of Proposition 3.5) in the proof of absence of non-zero solutions of the homogeneous equation tn = −
4π √ 4πα0 + ωn + s
αk tn+k ,
k∈Z k=0,−n
where s ∈ [0, ω). Let us suppose that there exists a non-zero solution {Tn } ∈ 2 (Z). Multiplying both sides of the equation by Tn∗ , one has √ ωn + s |Tn |2 = −4π Tn∗ αk−n Tk . (5.3) n∈Z n=0
n,k∈Z n,k=0
Since the right hand side is real, Tn = 0, ∀n < 0. Moreover, fixing T0 = 0 and setting Tn e−iωnt T (t) ≡ n∈Z
it follows that −4π
Tn∗ αk−n Tk = −4π T (t), α(t)T (t)
n,k∈Z
L2 ([0,T ])
≤0
because α(t) ≥ 0, ∀t ∈ [0, T ], but the left hand side of (5.3) is positive and then Tn = 0, ∀n ∈ Z. 6. Conclusions and Perspectives In Sects. 3, 4 and 5 we have proved that, under the genericity condition on α(t), the system defined in Sect. 2 shows asymptotic complete ionization, irrespective of its frequency. If inf(α(t)) < 0, the genericity condition may be a necessary condition to have complete ionization: for example, in one dimension, it is possible to exhibit (see [4]) explicit functions α(t) for which the genericity condition fails8 and the ionization is not complete. On the other hand, also in one dimension, it is not known whether the 8 A simple example of α(t), for which the genericity condition is not satisfied is the geometric series, αn = λ|n| for some λ < 1.
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condition is necessary. It would be interesting to check if non generic α(t) gives rise to asymptotic partial ionization in three dimensions. A possible way to investigate this problem is the analysis of the discrete spectrum of the Floquet operator. If one can find an explicit relation between existence of eigenvalues of the Floquet operator and the genericity condition, it would be probably easy to check if the condition is truly necessary. On the other hand, as we expected, if α(t) is positive at any time, no further condition on α(t) is required to prove complete ionization. Two interesting future applications of these methods can be the problem of complete ionization for moving point interactions and for N time-dependent point interactions. Indeed there are simple examples in which asymptotic complete ionization occurs also for moving sources (see [3]). Acknowledgement. M.C. is very grateful to Prof. Ludwik Dabrowski and the INTAS Research Project nr. 00-257 of European Community, “Spectral Problems for Schr¨odinger-Type Operators”, for the support.
References 1. Albeverio, S.A., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. New York: Springer-Verlag, 1988 2. Berezin, F.A., Faddeev, L.D.: A Remark on Schr¨odinger Equation with a Singular Potential. Sov. Math. Dokl. 2, 372–375 (1961) 3. Correggi, M., Dell’Antonio, G.F.: Rotating Singular Perturbations of the Laplacian. Ann. H. Poincar´e 5, 773–808 (2004) 4. Costin, O., Costin, R.D., Lebowitz, J.L., Rokhlenko, A.: Evolution of a Model Quantum System under Time Periodic Forcing: Conditions for Complete Ionization. Commun. Math. Phys. 221, 1–26 (2001) 5. Costin, O., Lebowitz, J.L., Rokhlenko, A.: Decay versus Survival of a Localized State Subjected to Harmonic Forcing: Exact Results. J. Phys. A: Math. Gen. 35, 8943–8951 (2002) 6. Costin, O., Lebowitz, J.L., Rokhlenko, A.: Exact Results for the Ionization of a Model Quantum System. J. Phys. A: Math. Gen. 33, 6311–6319 (2000) 7. Costin, O., Costin, R.D., Lebowitz, J.L.: Transition to the Continuum of a Particle in Time-Periodic Potentials. In: Advances in Differential Equations and Mathematical Physics, Birmingham 2002, Contemp. Math. 327, Providence, RI: AMS, 2003, pp. 75–86 8. Costin, O., Lebowitz, J.L., Rokhlenko, A.: Ionization of a Model Atom: Exact Results and Connection with Experiment. http://arxiv.org/abs/physics/9905038, 1999 9. Dell’Antonio, G.F., Figari, R., Teta, A.: Schr¨odinger Equation with Moving Point Interactions in Three Dimensions. In: Stochastic Processes, Physics and Geometry: New Interplays, Leipzig 1999, CMS Conference Proceedings 28, Providence, RI: AMS, 2000, pp. 99–113 10. Dell’Antonio, G.F.: Point Interactions. In: Mathematical Physics in Mathematics and Physics, Siena 2000, Fields Institute Communications 30, Providence, RI: AMS, 2001, pp. 139–150 11. Enss, V., Veselic, K.: Bound States and Propagating States for Time-dependent Hamiltonians. Ann. Inst. H. Poincar´e A 39, 159–191 (1983) 12. Figari, R.: Time Dependent and Non Linear Point Interactions. In: Proceedings of Mathematical Physics and Stochastic Analysis, Lisbon 1998, New York: World Scientific Publisher, 2000, pp. 184–197 13. Graffi, S., Grecchi, V., Silverstone, H.J.: Resonances and Convergence of Perturbative Theory for N-body Atomic Systems in External AC-electric Field. Ann. Inst. H. Poincar´e A 42, 215–234 (1985) 14. Howland, J.S.: Stationary Scattering Theory for Time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974) 15. Howland, J.S.: Scattering Theory for Hamiltonians Periodic in Time. Indiana Univ. Math. J. 28(3), 471–494 (1979) 16. Porter, D., Stirling, D.S.G.: Integral Equations. Cambridge: Cambridge University Press, 1990 17. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol.I: Functional Analysis. San Diego: Academic Press, 1975 18. Sayapova, M.R., Yafaev, D.R.: The Evolution Operator for Time-dependent Potentials of Zero Radius. Proc. Stek. Inst. Math. 2, 173–180 (1984)
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19. Yafaev, D.R.: Scattering Theory for Time-dependent Zero-range Potentials. Ann. Inst. H. Poincar´e A 40, 343–359 (1984) 20. Yajima, K., Kitada, H.: Bound States and Scattering States for Time Periodic Hamiltonians. Ann. Inst. H. Poincar´e A 39, 145–157 (1983) Communicated by B. Simon
Commun. Math. Phys. 257, 193–225 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1322-9
Communications in
Mathematical Physics
Noncommutative Spectral Invariants and Black Hole Entropy Yasuyuki Kawahigashi1, , Roberto Longo2, 1
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan. E-mail: [email protected] 2 Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy. E-mail: [email protected] Received: 11 May 2004 / Accepted: 15 October 2004 Published online: 15 March 2005 – © Springer-Verlag 2005
Dedicated to Richard V. Kadison on the occasion of his eightieth birthday Abstract: We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the “heat kernel” semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log µA , where µA is the global index of A, and the second spectral invariant is again proportional to c. We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S 1 and we get the same value proportional to c. We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log µA with a first order correction defined by means of the relative entropy associated with canonical states. By considering a class of black holes with an associated conformal quantum field theory on the horizon, and relying on arguments in the literature, we indicate a possible way to link the noncommutative area with the Bekenstein-Hawking classical area description of entropy. 1. Introduction This paper essentially deals with chiral conformal Quantum Field Theory, but our motivations primarily concern black hole thermodynamics; the basic link to this subject is
Supported in part by JSPS. Supported in part by GNAMPA and MIUR.
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through QFT on a curved spacetime and the idea, that has appeared from different and independent viewpoints in recent literature, that the restriction of the quantum field to the black hole horizon should give rise to a conformal QFT. Combined with the well known Bekenstein interpretation of the area of the horizon as proportional to the black hole entropy, this suggests that a geometric definition of the entropy of conformal QFT should play a relevant rˆole in black hole thermodynamics. To this end we shall define an intrinsic entropy associated to a conformal QFT, with a noncommutative geometrical point of view. We will regard a local conformal net as a noncommutative manifold or, more precisely, a QFT manifold (i.e. a noncommutative manifold with infinitely many degrees of freedom) and shall be guided in our analysis by the classical equivalent, most importantly from Weyl’s asymptotic for the trace of the heat kernel. One could say that in our framework back reaction effects of the quantum fields on the classical spacetime are negligible, but do affect the geometry of the associated noncommutative manifold. Our paper is organized as follows: • Here below we recall a number of ideas about black hole physics that have motivated our work, yet we refer to the literature (see e.g. [54]) for basics facts on black hole thermodynamics as Hawking effect, generalized second low, etc. • We then recall Weyl’s theorem that motivates our “log-ellipticity” assumption on the conformal Hamiltonian, i.e. on the asymptotic of logarithmic of the characters (elementary motivations are contained in Appendix B). This assumption holds in all computed cases. We shall show that it holds for all modular local conformal nets, namely nets with the usual rational behavior (see Sect. 3.2) and it turns out to hold in particular in all models with central charge less than one, that are classified in [30, 31]. Indeed one has the asymptotic formula for a modular net A, log Tr(e−2πtL0,ρ ) ∼
πc 1 πc d(ρ) − + log √ t, 12 t µA 12
as t → 0+ ,
(1)
where c is the central charge, L0,ρ and d(ρ) are the conformal Hamiltonian and the DHR 2dimension of the representation ρ, and µA is equal to the global index i d(ρi ) , the sum of the indices of all DHR charges [32, 39] (see Sect. 3.2). • Our basic object is a local conformal net A of von Neumann algebras, namely the family of local operator algebras maximally generated by smeared fields (basic notions can be found in Appendix A); this is our noncommutative manifold and we use (temporarily) the log-ellipticity/modularity assumption for our analysis. In analogy with Weyl’s theorem we define the noncommutative geometric spectral invariants {ai } of a conformal net (the coefficients in the above asymptotic (1,13)), in particular the noncommutative area and the noncommutative Euler characteristic. Indeed as we are in the QFT setting (thus with infinitely many degrees of freedom) log Tr(e−2πtL0 ), rather than Tr(e−2πtL0 ), provides the asymptotic of the corresponding finite-dimensional system, see Appendix B. From the Physics viewpoint, log Tr(e−2πtL0 ) counts logarithmically the number of possible states and so determines the microscopic entropy SA of the system, therefore we put SA ≡ a0 . The following table summarizes the value and the meaning of the spectral invariants (up to proportionality constants):
Noncommutative Spectral Invariants and Black Hole Entropy Invariant
Value
a0 a1 a2
π c/12 − 21 log µA −π c/12
195
Geometry
Physics
Noncommutative area Noncommmutative Euler characteristic 2nd spectral invariant
Entropy 1st order entropy 2nd order entropy
Note that a2 = −a0 , that is a consequence of the modular symmetry. The analog of the Kac-Wakimoto formula [36], and more generally the quantum index formula in [38], can now be read as an expression that the incremental free energy (adding/removing DHR charges [17]) is proportional to the increment of the noncommutative Euler characteristic (Sect. 3.3). • We shall show that, for a conformal net on the two-dimensional Minkowski spacetime, an expansion analog to (1) holds, where a0 duplicates. At this point we look for a direct connection with black hole thermodynamics. In the paper [11] (following [49]) on black holes one finds computations that fit well with our results. There c/12 = A/8π, so one immediately gets that SA has the Bekenstein behavior SA = A/4 , where A is the classical area of the black hole horizon. • We then provide a general analysis where we do not any longer use the modularity assumption. We first recall how the n-cover Diff (n) (S 1 ) of Diff(S 1 ) acts on S 1 , see [39]. The generator of the corresponding rotation one-parameter group is viewed as a conformal Hamiltonian associated with a discretization of S 1 , namely to a partition of S 1 in n intervals, where n is then supposed to tend to infinity. If one subtracts from the corresponding entropy (logarithm of partition function) the naive entropy associated with 1/n times the original conformal Hamiltonian, the resulting entropy should take into account the noncommutative geometrical complexity. We thus give in this way a general definition of mean free energy and it turns out immediately that Fmean = πc/12 , that agrees with the above found value for the entropy a0 , hence again Fmean = A/4 in the above setting. • At this point we get in the second part of the paper, where we study the “local” version of the above structure, namely we consider the operator algebra associated with a given interval and the associated proper dynamics with a one-parameter group of special conformal transformations. We consider the generators of this “dilatation” group in Diff(S 1 ) and in Diff (n) (S 1 ) as Hamiltonians and we attempt to compute the associated noncommutative spectral invariants. Only conformal symmetries and the split property play a rˆole here and results are very general. We then extend to the general model independent setting a formula by Schroer and Wiesbrock [48] for the Tomita-Takesaki modular group of the von Neumann algebra associated with n separated intervals; in other words we prove the KMS thermal equilibrium property, for the above proper dynamics associated with the discretization of S 1 , with respect to a canonical state, in any representation. This is one of our main tools for the sequel. • With this proper Hamiltonian, in analogy with the previous analysis, we define the partition function Zn associated with this discretization of S 1 with n-intervals and then the µ-free energy Fmean,µ as the limn→∞ −β −1 log Zn (β)/n at inverse equilibrium temperature β (Hawking temperature). It turns out that, in any irreducible representation,
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Fmean,µ =
1 log µA , 2
where µA is the µ-index of the net, namely the Jones index of the 2-interval inclusion of von Neumann algebras in the vacuum sector [32] (Sect. 3.2). Pursuing the above analogy we interpret the first noncommutative local spectral invariants. It turns out that the 0th invariant a0,µ , equal by definition to the proper noncommutative area, is proportional to the mean entropy. The first spectral invariant a1,µ , equal by definition to the proper noncommutative Euler characteristic, turns out to be proportional to the proper mean entropy Smean,µ . (Locally the µ-index seems to play the rˆole of the central charge globally, but we have no definite interpretation of this fact.) • Our mathematical methods concern Jones’ index [27], as extended by Kosaki [33], and Connes-Haagerup noncommutative measure theory, see [52]. We have put our mathematical results in Appendix C, in order not to interrupt the main theme of the paper. A quick introduction to Operator Algebras and Conformal Field Theory can be found in [29]. 2. On Black Hole Entropy We now recall a few motivational items concerning black hole physics. The holographic principle. [25, 51]. The celebrated Bekenstein formula [3] for the entropy of a black hole is S = αA, where A is the area of the black hole horizon and α is a constant. This formula was initially motivated by consistency arguments and the area theorem. One of the most surprising facts is that it sets the entropy to be proportional to the area, rather than to the volume, as an intuitive picture of the entropy as logarithmic counting of the number of possible states would suggest. This dimensional reduction has more recently led to the formulation of the holographic principle according to which, in a theory combining quantum theory and gravity, the degrees of freedom of a three dimensional world can be stored in a two dimensional projection. One of the arguments is that “one can’t hide behind a black hole”: if a black hole projects itself on a screen, due to gravity a second black hole can’t eclipse its image on the screen [51]. Hawking temperature. Fixing the proportionality constant. Let’s recall how the proportionality constant can be fixed as α = 1/4 by considering quantum effects (cf. [53]). As shown by Hawking, a black hole emits a thermal radiation with inverse temperature β=
2π , κ
where κ is the surface gravity. Let’s consider the Schwarzschild spacetime with radius R, 1 thus describing a black hole of mass M = 2R. In this case κ = 4M , thus β = 8π M. As S = αA = α4πR 2 = α16π M 2
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we have dS = α32πMdM. On the other hand by the generalized second principle of thermodynamics dS = βdH = βdM , where H = M is the energy, so β = 8πM = α32π M yielding α = 1/4. Limit of information. Discretization of the horizon [4]. Consider the horizon to be made by cells of area ∼ 2 , where is the Planck length. Thus A = n2 . Now say that each cell has k degrees of freedom: in the simplest example each cell is occupied by a particle with spin up/spin down and so k = 2. The total number of degrees of freedom is then Degrees of freedom = k n ;
(2)
thus A log k, (3) 2 where C is a constant, namely the entropy is proportional to the area A of the black hole. It follows that the increment of entropy by adding a particle to the black hole Entropy = Cn log k = C
dS = C log k
(4)
is proportional to the logarithm of an integer. More generally if there are distinct particles p1 , p2 , . . . ps and pi has ki degrees of freedom we have Degrees of freedom = k1n1 k2n2 · · · ksns , where n = n1 + n2 + . . . ns , so Entropy = C log k1n1 k2n2 · · · ksns = C
ni log ki .
(5)
(6)
i
The conformal horizon of a black hole. The horizon of a black hole is the boundary of the no escape region of the spacetime where signals can enter, but cannot get out. There is no particular physical phenomena occurring on the horizon, an observer can cross it without feeling anything, yet it is a codimension one submanifold where certain parameters (coordinates) pick critical values. For this reason it is thus natural to expect the horizon to exhibit further symmetries acquainted at these critical values. This point, related to the above mentioned holographic principle, is well expressed in the holography that holds in the anti-de Sitter spacetime [40]. Here the algebraic approach gives a natural “coordinate free” description [44]. More recently a general algebraic holography has been realized in the two-dimensional de Sitter spacetime by means of local conformal (pseudo)-nets of von Neumann algebras on S 1 [21]. There is an apparent conflict between the discretization of the boundary and conformal invariance: our point of view is that the conformal symmetries that respect to the discretization are the physically relevant ones. One should think of conformal QFT on the boundary as a noncommutative manifold, and we shall soon be back on this point. The corresponding structure will be explained later on.
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Entropy from conformal boundary. This point of view has emerged in recent years in different works as in [49, 11, 2] where conformal symmetries on the horizon are used to compute black hole entropy. For example, in reference [11] by Carlip the black hole is described, in particular, by a spacetime with a (local) Killing horizon; a natural set of boundary conditions leads to a representation of the Virasoro algebra with central charge c and it is argued that, in normalized units, c A = , 12 8π
(7)
where A is the area of a cross section of the horizon (the black hole area). One then uses a heuristic formula derived with a certain assumption by Cardy ρ(λ) ∼ exp 2π
1 1 6 c(λ − 24 c)
as λ → +∞
(8)
on the number of states ρ(λ) corresponding to the eigenvalue λ of the (two-dimensional) conformal Hamiltonian. One computes the boundary term of the energy (that turns out to be equal to = A/8π), inserts this and the value of c in Eq. (8) and gets the expected Bekenstein behavior log ρ ∼
A . 4
Operator algebras and conformal boundary. Quantum index theorem. Recall now the work in [22, 37] in the context of black holes described by a curved spacetime with a bifurcate Killing horizon. A is a conformal net arising on the horizon. By applying a general theorem by Wiesbrock, A is a M¨obius covariant net (cf. [50, 37, 38, 47]); moreover A is expected to be diffeomorphism covariant and the diffeomorphism symmetry uniquely determined (see [12]); for example this is the case when the quantum field is free, as A is then isomorphic to the net associated with the U (1)-current algebra, see [22] (this fact has been noticed again in [41]). We thus assume A to be diffeomorphism covariant. In [36–38] one obtained a general, model independent formula for a black hole with a bifurcate Killing horizon (assuming the KMS property for geodesic observers): dF =
2π log d(ρ) − log d(σ ) , κ
(9)
where dF is the incremental free energy by adding/removing DHR charges ρ, σ localizable in bounded regions ([17]), κ/2π is the Hawking temperature with κ the surface gravity, d(ρ) is the Doplicher-Haag-Roberts statistical dimension of ρ, that turns out to be equal to the square root of the Jones index of ρ [34]. Recall that, in a n-dimensional spacetime, n ≥ 3, we have d(ρ) ∈ N∪∞. The above formula holds also for finitely many charges, and we regard (3) as a physical description of (9). It can be read as a quantum index theorem (or, more appropriately, “QFT index theorem” as it concerns infinitely many degrees of freedom) where the quantum Fredholm index log d(ρ) − log d(σ ) is expressed in terms of dF and the geometric quantity κ. A good illustration of this point is provided by the topological sectors in [39].
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3. QFT, Heat Kernel Asymptotic and Entropy 3.1. Weyl’s theorem and ellipticity. Let M be a compact oriented Riemann manifold and
the Laplace operator on L2 (M). The eigenvalues of M can be thought as “resonant frequencies” of M and capture most of the geometry of M [28]. At the root of this analysis is the famous Weyl’s theorem on the asymptotic density distribution of such eigenvalues. This can be stated as an asymptotic formula for the heat kernel, see [46]. One has the following asymptotic expansion as t → 0+ : Tr(e−t ) ∼
1 (a0 + a1 t + · · · ) (4πt)n/2
(10)
and thus, by Tauberian theorems (see [5]), the asymptotic formula as λ → +∞ N (λ) ∼
vol(M) λn/2 (4π)n/2 ((n/2) + 1)
for the number N (λ) of eigenvalues of less than λ, where is Euler Gamma-function. In (10) the spectral invariants n and a0 , a1 , . . . encode geometric information and in particular n = dim(M) and 1 a0 = vol(M), a1 = κ(m)dvol(m), 6 M where κ is the scalar curvature, thus in particular if n = 2 then a1 is proportional to the 1 Euler characteristic equal 2π M κ(m)dvol(m) by Gauss-Bonnet theorem. Motivated by the Weyl asymptotic (10), having in mind a “second quantized” Hamiltonian (see Sect. B, in particular Lemma 24), we give the following definition to capture the asymptotic associated with the (here undefined) “one-particle Hamiltonian”. A positive linear operator H on a Hilbert space is log-elliptic if there exists n > 0 and ai ∈ R, a0 = 0, such that log Tr(e−tH ) ∼
1 (a0 + a1 t + · · · ) as t → 0+ . t n/2
(11)
Then n = −2 lim
t→0+
log log Tr(e−tH ) log t
is called the dimension of H and ai ≡ ai (H ) the i th spectral invariant of H . The following is obvious. Lemma 1. Let H , H be log-elliptic positive linear operators with dimension n and n
and spectral invariants ai and ai . If lim
t→0+
Tr(e−tH )
= λ = 0 Tr(e−tH )
exists, then n = n and ai = ai , i = 0, 1, 2, . . . , m − 1, m ≡ n/2 ; if n/2 is an integer
. then log λ = am − am
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Proof. We have Tr(e−tH ) −tH −tH
) − log Tr(e ) log Tr(e log λ = lim log = lim
Tr(e−tH ) t→0+ t→0+ 1 1 2
2 = lim (a + a t + a t + . . . ) − (a + a t + a t + . . . ) (12) 0 1 2
1 2 t n /2 0 t→0+ t n/2 which is possible only in the stated case.
3.2. Spectral invariants associated with L0 . The asymptotic of the character Tr(e−2πtL0 ) as t → 0+ is known for an irreducible representation of the Virasoro algebra [53], but is unknown for a general reducible representation, in particular for the representation associated with an arbitrary local conformal net. Cardy has provided an argument based on modular invariance that implies 1 as t → 0+ , t where the constant depends on the central charge c only. Motivated by Weyl’s theorem and the above expansion, we shall define a local conformal net A to be two-dimensional log-elliptic (or QFT-elliptic) if its conformal Hamiltonian L0 is log-elliptic with dimension 2, see Sect. 3.1, namely log Tr(e−2πtL0 ) ∼ const.
1 (13) (a0 + a1 t + · · · ) as t → 0+ , t log-ellipticity is essentially the nuclearity condition of Buchholz and Wichmann [9] (and we fix the dimension). We shall then regard A as a 2-dimensional noncommutative manifold, where L0 corresponds to the Laplacian and the spectral invariants of L0 are noncommutative geometric invariants for A. In particular a0 ≡ a0 (2πL0 ) is 1/4π times the noncommutative area of A and 12a1 is the noncommutative Euler characteristic of A. 1 Of course a0 , a1 , . . . have a priori no classical geometric interpretation, but are defined in analogy with classical invariants. We now explain how to obtain a more precise form of the asymptotic (13) under a general condition. Let A be a completely rational local conformal field net on S 1 . For a DHR sector ρ, we consider the specialized character χρ (τ ) for complex numbers τ with Im τ > 0 as follows: χρ (τ ) = Tr e2πiτ (L0,ρ −c/24) . log Tr(e−2πtL0 ) ∼
Here the operator L0,ρ is the conformal Hamiltonian in the representation ρ and c is the central charge. We assume that the above Trace converges, which in particular means each eigenspace of L0,ρ is finite dimensional. On one hand, it is known in many cases that we have an action of SL(2, Z) on the linear span of these specialized characters through a change of variables τ as follows: χ Sρ,ν χν (τ ), χρ (−1/τ ) = ν (14) χ χρ (τ + 1) = Tρ,ν χν (τ ). ν 1
For simplicity we do not put a factor 1/4π in defining the asymptotic (13).
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On the other hand, we have a unitary representation of the group SL(2, Z) on the space spanned by the sector ρ’s arising from the nondegenerate braiding as in Rehren [43], in particular we have the associated matrices (Sρ,ν ) and (Tρ,ν ). It has been conjectured, e.g. Fr¨ohlich-Gabbiani [19, p. 625], that these two representations coincide, that is, we have S χ = S, T χ = T . Note that we always have T χ = T by the spin-statistics theorem [20], so in order to verify these identities, it is enough to show that the fusion rules dictated by S χ and the fusion rules dictated by composition of DHR-sectors coincide. Such identification of the two fusion rules have been verified in many examples including all local conformal nets with central charge less than 1 classified in [30]. Also note that if these two representations of SL(2, Z) coincide, we have the following Kac-Wakimoto formula, as explained in [19, p. 626], χ χ Sρ,0 Sρ,ν χν (τ ) χρ (τ ) Sρ,0 = lim d(ρ) = = χ = lim ν χ . (15) τ →0 τ →i∞ S0,0 χ0 (τ ) S0,0 S χ (τ ) ν 0,ν ν Here we denote the vacuum sector by 0 and d(ρ) is the statistical dimension of ρ. (Note that we have hρ > 0 for ρ = 0, where hρ is the lowest eigenvalue of the operator L0,ρ , see Lemma 21.) We shall say that A is modular if the µ-index µA < ∞ (see Sect. 6.1), the modular symmetries (14) hold (in particular the characters are defined, namely Tr(e−tL0,ρ ) < ∞) and the above two representations of SL(2, Z) are identical. Note that a modular net is completely rational. Modularity holds in all computed rational cases, cf. [55]. The SU (N )k nets and the Virasoro nets Vir c with c < 1 are both modular. We expect all local conformal completely rational nets to be modular (see [26] for results of similar kind). Furthermore, we have the following. Proposition 2. Let A be a modular local conformal net and B an irreducible extension of A. Then B is also modular. Proof. Since A is completely rational, the extension has finite index and B is also completely rational. We denote the S-matrices for A and B arising from the braid˜ respectively. For irreducible DHR sectors ρ and σ of A ing as in [43] by S and S, and B, respectively, we put bσ,ρ = dim(αρ , σ ), where αρ is α-induction. This bσ,ρ is equal of ρ in the representation σ restricted to A. Then we have to the multiplicity ˜
b = b S
σ σ,σ σ ,ρ ρ σ,ρ Sρ ,ρ by [6, Theorem 6.5]. We now have χσ (−1/τ ) =
bσ,ρ χρ (−1/τ )
ρ
=
bσ,ρ Sρ,ρ χρ (τ )
ρ,ρ
=
S˜σ,σ bσ ,ρ χρ (τ )
σ ,ρ
=
S˜σ,σ χσ (τ ).
σ
This shows that the matrix S˜ arising from the braiding for B also gives a transformation matrix for the characters.
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Proposition 3. Assume that A is modular. Then the following asymptotic formula holds: πc 1 1 πc − log µA − t 12 t 2 12
log Tr(e−2πtL0 ) ∼ Proof. We first have
Tr(e−2πtL0 ) = e−cπt/12
as t → 0+ .
S0,ν ecπ/(12t) Tr(e−2πL0,ν /t ).
ν
Then in this finite summation, the terms for ν = 0 are exponentially smaller than the term for ν = 0. This gives Tr(e−2πtL0 ) ∼ S00 e− 12 (t−1/t) , πc
therefore log Tr(e−2πtL0 ) ∼ − −1/2
and we know that S00 = µA
cπ cπ 1 t + log S00 + , 12 12 t
(e.g. [43]), so we get the above statement.
In particular, in the case c < 1, two-dimensional log-ellipticity can be proved for all local conformal nets. We give also an independent proof of this corollary as follows. Corollary 4. Let A be a local conformal net with c < 1. Then A is two-dimensional log-elliptic with noncommutative area a0 = 2πc/24, thus log Tr(e−2πtL0 ) ∼
c 2π 24 t
as t → 0+ .
Proof. The Virasoro net Vir c with a central charge c < 1 is completely rational and A is a finite index extension of Vir c [30]. Hence the conformal Hamiltonian L0 of A is a finite direct sum of conformal Hamiltonians associated with irreducible representations of Vir c . As the stated asymptotic is valid for all these conformal Hamiltonians [53, Prop. 6.14], the proposition holds true. Corollary 5. Let A be modular and ρ a representation of A. The following asymptotic formula holds: log Tr(e−2πtL0,ρ ) ∼
d(ρ)2 πc 1 1 πc + log t − 12 t 2 µA 12
as t → 0+ .
Proof. We can assume d(ρ) < ∞ as otherwise both members of the asymptotic equality are infinite. By using Prop. 3 and the Kac-Wakimoto formula (15,17), we have −2πtL0,ρ ) −2πtL0,ρ −2πtL0 Tr(e log Tr(e ) = log Tr(e ) Tr(e−2πtL0 ) Tr(e−2πtL0,ρ ) Tr(e−2πtL0 ) πc 1 1 πc ∼ − log µA − t + log d(ρ), 12 t 2 12
= log Tr(e−2πtL0 ) + log
hence the corollary follows.
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We note explicitly that the information on the normalized index is contained in the spectral density of the Hamiltonian: log d(ρ) −
d 1 log µA = lim t log Tr(e−tL0,ρ ) . + 2 t→0 dt
Because of the above formula we conjecture that a local confomal net A is completely rational iff lim
t→0+
d t log Tr(e−tL0 ) < ∞. dt
Recall now the following particular case of Kohlbecker’s Tauberian theorem [5, Th. 4.12.1]. Let m be a Borel measure on [0, ∞) finite on compact sets. The logarithm of the Laplace transform has the asymptotic behavior 1 log e−tλ dm(λ) ∼ C as t → 0+ , t C > 0, if and only if √ log m[0, λ] ∼ 2 Cλ,
as λ → +∞ .
(16)
As a further corollary, we then have an asymptotic formula which is, in part, a version of Cardy’s formula (8). Notice however that formula (8) concerns CFT on a two-dimensional spacetime, while we deal with conformal nets on S 1 . Corollary 6. Let A be a modular local conformal net on S 1 and ρ an irreducible representation of A. Then
c λ as λ → ∞, log N (λ) ∼ 2π 6 where N (λ) is the number of eigenvalues (with multiplicity) of L0,ρ that are ≤ λ. Proof. By Cor. 4 we have log Tr(e−tL0,ρ ) ∼ C/t with C = π 2 c/6. As Tr(e−tL0,ρ ) = e−tλ dm(λ), √ √ where m[0, λ] = N (λ), (16) reads log N (λ) ∼ 2 2π 2 c/12 λ = 2π cλ/6.
From the physics viewpoint it is natural to define SA , the entropy of A, as the leading coefficient of the expansion (13) of log Tr(e−2πtL0 ), thus a0 = SA , a1 , a2 , · · · = higher order corrections to SA . Note that, by definition, the entropy is proportional to the noncommutative area: it is just a matter of reading the same formula from different points of view.
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3.3. The incremental free energy in [36] (increment of the first spectral invariant). Let A be a local conformal net and ρ, σ a DHR representation of A (see Sect. A). The above mentioned Kac-Wakimoto formula lim
t→0+
Tr(e−tL0,ρ ) d(ρ) = d(σ ) Tr(e−tL0,σ )
(17)
has been tested in wide generality and always holds true, see [55], and we have just seen to hold true if A is modular. Proposition 7. If A is modular, then log d(ρ) − log d(σ ) = a1 (2πL0,ρ ) − a1 (2πL0,σ ) ≡
1 12 (χρ
− χσ ),
where χρ − χσ is the increment of the noncommutative Euler characteristic by adding the charge ρ and removing the charge σ . Proof. This is an immediate corollary of Prop. 5.
Recall now the work in [22, 37] in the context of black holes described by a curved spacetime with a bifurcate Killing horizon. There A is a local conformal net canonically arising on the horizon. According to the general analysis (by using Wiesbrock’s theorem) A is a M¨obius covariant net, but A is expected to be diffeomorphism covariant too [12]; for example this is the case when the quantum field is free, as A is then isomorphic to the net associated with the U (1)-current algebra [22] (see also [41]). The incremental free energy dF by adding the charge ρ and removing the charge σ (in the Hartle-Hawking state) in [36] or, more generally, its symmetrization, see [38, Thm. 5.4], is defined and turns out to be given by 2π dF = β log d(ρ) − log d(σ ) = log d(ρ) − log d(σ ) , κ
(18)
where κ is the surface gravity and β ≡ 2π/κ is the Hawking temperature. We thus assume A to be diffeomorphism covariant and that Prop. 7 holds. Recall that, in higher dimensional spacetimes, d(ρ) ∈ N ∪ ∞ [17]. We then have: Corollary 8. With the above assumptions, the incremental free energy by adding the DHR charge ρ and removing the charge σ is proportional to the increment of the noncommutative Euler characteristic π dF = (19) χσ − χ ρ . 6κ Adding a charge is proportional to the logarithm of an integer. Proof. The proof is immediate from the above discussion.
The above formulas (18,19) are consistent with the interpretation of the entropy by logarithmic counting states and the fact that it is proportional to an integer as in Eq. (3). Compared with the work [36], the above corollary expresses the incremental free energy by a true difference of global entropies log Tr(e−tL0,ρ ) and log Tr(e−tL0,σ ) by Prop. 7.
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3.4. Relation to black hole entropy. I. A microscopic derivation of black hole entropy and its relation to conformal symmetries and central charge is discussed in [49]. The potentiality of our discussion in relation to black hole entropy and Bekenstein classical area description is well exemplified if one relies on the reference [11] recalled in Sect. 2. Yet we use here only the value of the central charge (Eq. (7)) and not Cardy’s formula nor the boundary term of the energy. We shall make here the assumption that the associated local conformal net A is modular. (Later we shall introduce the mean free energy and put it in relation to Bekenstein entropy, on the same lines, without the modularity assumption.) Corollary 9. For a black hole in the above class [11], we have SA = A/4, where A is the area of the black hole horizon. Proof. Immediate from the relation c/12 = A/8π (7) and the value SA = 2π c/12 of the entropy for modular nets on the two-dimensional Minkowski spacetime. We have therefore the picture in the following diagram: geometry
physics
Entropy −−−−→
a0 ←−−−−− 4π · Noncommutative area modular nets
2πc/12 black hole models A/4 4. Discretization and Conformal Invariance There is an apparent conflict in regarding the horizon of a black hole both having a discrete essence and a conformal group of symmetries. In the sequel we take simultaneously account of both pictures by considering the n-cover Diff (n) (S 1 ) of Diff(S 1 ) acting on S 1 and respecting the cell partitioning of S 1 . Thus the conformal Hamiltonian becomes the generator of the rotation group for the unitary action of Diff (n) (S 1 ). We then consider mean quantities, as entropy, as n tends to infinity. Note that in the sequel of this paper we shall not any longer need the modularity or log-ellipticity assumptions. 4.1. The action of the n-cover of Diff(S 1 ). We recall now some facts on Diff (n) (S 1 ) and its canonical embedding into Diff(S 1 ), see [39]. The Virasoro algebra is the infinite dimensional Lie algebra generated by elements {Ln | n ∈ Z} and c with relations [Lm , Ln ] = (m − n)Lm+n +
c (m3 − m)δm,−n 12
(20)
and [Ln , c] = 0. It is the (complexification of) the unique, non-trivial one-dimensional central extension of the Lie algebra of Vect (S 1 ).
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The elements L−1 , L0 , L1 of the Virasoro algebra are clearly a basis of s(2, C). The Virasoro algebra contains infinitely many further copies of s(2, C): for every fixed (n) (n) (n) n > 0 we get a copy generated by the elements L−1 , L0 , L1 , where (n)
L±1 ≡ n1 L±n , (n) L0
≡ n1 L0 +
c 24
(21)
(n2 −1) n
(22)
.
We have indeed (n)
(n)
(n)
(n)
[L1 , L−1 ] = 2L0 ,
(n)
(n)
[L±1 , L0 ] = ±L±1
(23)
that are the relations for the usual generators in s(2, C). It follows that, setting for a fixed n > 0, L(n) m ≡
1 Lnm , n
m = 0 ,
(24)
(n)
and L0 as in (22), the map
(n)
Lm → Lm c → nc ,
gives an embedding of the Virasoro algebra into itself. There corresponds an embedding of Diff (n) (S 1 ), the n-cover of Diff(S 1 ), into Diff(S 1 ) as stated in the following. Proposition 10 ([39]). There is a unique continuous isomorphism M (n) of Diff (n) (S 1 ) into Diff(S 1 ) such that for all g ∈ Diff (n) (S 1 ) the following diagram commutes: (n)
Mg
S 1 −−−−→ zn
S1 n
z
(25)
Mg
S 1 −−−−→ S 1 (n)
i.e. Mg (z)n = Mg (zn ) for all z ∈ S 1 . Here g is the element of Diff(S 1 ) corresponding to g and Mg is the obvious action of g on S 1 . ¨ is the n-cover of Mob ¨ and M (n) restricts Mob ¨ (n) ≡ {g ∈ Diff (n) (S 1 ) : g ∈ Mob} (n) 1 to an embedding of Mob ¨ into Diff(S ). (n)
Clearly the embedding Mob ¨ (n) → Diff(S 1 ) corresponds to the embedding Lm → Lm , m = −1, 0, 1, of s(2, C) into the Virasoro algebra.
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4.2. The mean free energy (topological increment of the second spectral invariant). Let A be a local conformal net on S 1 (in any representation).√We divide S 1 into n equally n spaced cells, namely we consider the n-interval En ≡ S + , where S + is the upper semicircle. Each interval component Ik of En contains minimal information (as the cells of Planck length). There is a canonical evolution associated with En corresponding to the rotations on the full S 1 , namely the rescaled rotations R( n1 ϑ), giving rise to two (n) rescaled conformal Hamiltonians: one, Lˆ 0 ≡ n1 L0 , comes by purely rescaling the Hamiltonian, the other is the one associated with the representation U (n) of Diff (n) (S 1 ), (n) c (n2 −1) namely L0 = n1 L0 + 24 n , and takes care of “boundary effects”. The geometrical complexity should be encoded in the difference between the two terms. We define the free energy associated with the above partition of S 1 as the difference of the free energy associated by the corresponding partition functions at infinite temperature: ˆ (n)
(n)
Fn ≡ t −1 log Tr(e−t2πL0 ) − t −1 log Tr(e−t2π L0 ) (one could generalize the definition of Fn without the existence of characters, but we do c (n2 −1) not dwell on this point). Clearly Fn = 24 n 2π, hence we get the following model independent formula for the mean free energy associated to the “discretization of S 1 ”. Theorem 11. Let A be a local conformal net. We have c Fmean = 2π . 24 Proof. Obviously Fmean ≡ limn→∞ n1 Fn = 2πc/24.
(26)
Note that we clearly have the relation a2 (2πL0 ) − a2 (2π Lˆ 0 ) = Fn , (n)
(n)
thus also Fmean has a noncommutative geometrical meaning. Concerning a two-dimensional conformal QFT, both chiral components contribute to the topological entropy thus, assuming the central charge to be equal for both components, the physical topological entropy duplicates c Fmean = 2π ; (27) 12 we shall explain this point in Sect. 7. 4.3. Relation to black hole entropy. II. As noted, the derivation of the value Fmean = 2π c/12 is model independent and general; essentially it follows only by diffeomorphism invariance. As the value of Fmean coincides with the value of SA (for modular nets), we now have a link with the classical area restriction, just as in Sect. 3.4, without any modularity assumption on A. For a black hole as in Corollary 9, we have indeed Fmean = A/4, where A is the area of the black hole horizon. This is immediate from the relation c/12 = A/8π (7) and the found value Fmean = 2π c/12 of the two-dimensional free energy (27).
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5. The Modular Group of a n-Interval von Neumann Algebra Here we extend to the general model independent setting, and in an arbitrary representation, a formula (announced in [39]) for the modular group discussed by Schroer and Wiesbrock [48] in the context of the U (1)-current algebra √ local conformal net. Let E be a symmetric n-interval of S 1 , thus E ≡ n I for some I ∈ I, i.e. E = {z ∈ S 1 : zn ∈ I }. Let I0 , I1 , · · · In−1 be the n connected components of E; we may assume that Ik = R(2π k/n)I0 , where R is the rotation subgroup of Mob. ¨ Let A be a local conformal net on S 1 with the split property, in an irreducible representation. By the split property we have a natural isomorphism χE : A(E) ≡ A(I0 ) ∨ A(I1 ) ∨ · · · ∨ A(In−1 ) → A(I0 ) ⊗ A(I1 ) ⊗ · · · ⊗ A(In−1 ) . A product state ϕ is a state on A(E) of the form ϕ ≡ (ϕ0 ⊗ ϕ1 ⊗ · · · ⊗ ϕn−1 ) · χE , where ϕk is a normal faithful state on A(Ik ) and ϕk = ϕ0 · AdU (R(2kπ/n)) is called a rotation invariant product state. We now exhibit a modular group of A(E) having a geometrical meaning. Let k be the isomorphism between A(Ik ) and A(I ) associated with the function zn , namely k (x) ≡ U (hk )xU (hk )∗ ,
x ∈ A(Ik ),
where hk is any element of Diff(S 1 ) such that hk (z) = zn , z ∈ Ik , (by locality the definition of k is independent of the choice of hk ). Let ϕk be the state on A(Ik ) given by ϕk ≡ ωI · k , where ω is the vacuum state, and let ϕE be the product state on A(E) that restricts to ϕk on A(Ik ). Clearly ϕE is a rotation invariant product state. Theorem 12. Let A be a local conformal net in a irreducible √ representation and U the covariance unitary representation of Diff(S 1 ). With E = n I an n-interval as above, the canonical rotation invariant product state ϕE on A(E) has the modular group σ ϕE given by ϕE
σt
= AdU (n) (I (−2πt))A(E) ,
¨ (n) of the one-parameter subgroup of Mob ¨ of generalized where I is the lift to Mob dilatation associated with I (see Appendix A) and U (n) = U · M (n) is the unitary representation of Mob ¨ (n) associated with U . ϕ
Proof. Since both σt E and AdU (n) (I (−2πt))A(E) are tensor products of their restrictions to the components A(Ik ), by rotation invariance it suffices to prove the formula on each A(Ik ). We have ωI σt E A(Ik ) = σtω·k A(Ik ) = −1 k · σt · k ϕ
(n) = −1 (I (−2π t)) A(Ik ) . k · AdU (I (−2πt)) A(I ) ·k = AdU
(28)
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209
Corollary 13. In the above proposition, setting V (t) ≡ U (n) (I (−2π t)), we have: ϕ
ϕ
AdV (t) A(E) = σt E ,
AdV (−t) A(E ) = σt E . √ n Proof. The first equality has been already shown. Since E = I , to get the second (n) equality we just have to show that V (−t) = U (I (−2π t)), which is clearly the case since I (−t) = I (t). Note that the abstract results in Appendix C now apply. 6. Entropy and Global Index with the Proper Hamiltonian In this section we pursue the above point of view, but we replace the conformal Hamiltonian L0 with the “local” Hamiltonian K1 ≡ i(L1 − L−1 ) , the generator of the one-parameter dilatation unitary group associated with the upper semicircle S + (see Appendix A). With this dynamics, the restriction of the vacuum state satisfies the equilibrium condition at Hawking temperature and is natural to be considered, see e.g. [23, 54, 36, 38]. As above we will consider the corresponding dynamics for the action of Diff (n) (S 1 ) and compute noncommutative spectral invariants. It turns out the analysis below can be done in complete generality: it is only based on conformal invariance and the split property (recall that the latter follows automatically from the existence of characters). 6.1. µ-index. Let A be a local conformal net with the split property in the vacuum representation and E ⊂ S 1 a 2-interval, namely E and its complement E are the union of two proper intervals. The µ-index of A is defined as ˆ : A(E)], µA ≡ [A(E) ˆ where the brackets denote the Jones index and A(E) ≡ A(E ) . It turns out that µA does not depend on E and µA = d(ρi )2 i
sum over the indices of all irreducible DHR charges, namely µA coincides with the global index of A. More generally, if En is an n-interval, and in the representation ρ, we have ρ ˆ n ) : A(En )] = d(ρ)2 µn−1 . µA,n ≡ [A(E A
Note that the formula
µA = lim
n→∞
n
ˆ n ) : A(En )] [A(E
gives the µ-index in any irreducible representation. Indeed we have:
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Proposition 14. Let A be a split, local M¨obius covariant net in an irreducible representation ρ. Given an interval I , both µA and d(ρ) can be measured in I . Proof. Fix an interval I and divide I in 2n−1 contiguous intervals I1 < J1 < I2 < J2 < · · · < Jn−1 < In , where < denotes the counter-clockwise order. Then ∨ni=1 A(Ii ) ⊂
n−1 ∨i=1 A(Ji ) ∩ A(I ) is an n-interval inclusion, thus its index is equal to d(ρ)2 µn−1 A and we have 1
1 n 2 n−1 n n = µA , lim [ ∨n−1 i=1 A(Ji ) ∩ A(I ) : ∨i=1 A(Ii )] = lim d(ρ) µA
n→∞
n→∞
2 showing that µA can be detected within the interval I and so is the case also for d(ρ) =
n−1 n
∨n−1 i=1 A(Ji ) ∩ A(I ) : ∨i=1 A(Ii ) /µA (for instance with n = 2).
As is known, the central charge may also be measured locally, as it appears locally in the commutation relations with the stress-energy tensor. 6.2. µ-entropy and spectral invariants for the proper Hamiltonian. Let A be a local conformal net on S 1 with the split property in an irreducible representation ρ. Let I = S + be √ n the upper semicircle, E ≡ En = I the associated n-interval and Kn the infinitesimal generator of V (n) , where V (n) (t) = U (n) (I (−2πt)) as in Cor. 13. Note that (n)
(n)
Kn ≡ i(L1 − L−1 ) = ni (Ln − L−n ) . √ n The complement En of En is the n-interval En = I . Let ϕEn be the rotation-invariant product state on A(En ) defined in Prop. 12 and ξn ≡ ξEn a cyclic separating vector for A(En ) implementing ϕEn . Theorem 15. We have n−1
(e−2πKn ξn , ξn ) = d(ρ)µA2 , thus log(e−
2π i n (Ln −L−n )
ξn , ξn ) = =
n−1 2 n−1 2
log µA + log d(ρ) log( d(ρi )2 ) + log d(ρ). i
Proof. The unitary U (R(2π/n)) implements an isomorphism between A(En ) and ˆ n ) and A(E ˆ n ); moreover it maps ϕE to ϕEn and Kn to −Kn . A(En ), and between A(E n
Hence, if ξn is a cyclic and separating vector for A(En ) implementing the state ϕEn , we have (e−2πKn ξn , ξn ) = (e2πKn ξn , ξn ), thus by Cor. 29 ρ ˆ n ) : A(En )] = d(ρ)2 µn−1 . (e−2π Kn ξn , ξn )2 = µA,n ≡ [A(E A
Noncommutative Spectral Invariants and Black Hole Entropy
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If the µ-index is finite, we shall denote by ϕˆEn = ϕEn · εEn ˆ n ) obtained by extending ϕEn by the conditional expectation εEn : the state on A(E ˆ ˆ n ) is defined analogously. If µA = ∞ there A(En ) → A(En ). The state ϕˆEn on A(E ˆ n ) → A(En ) by the Haagerup theorem and exists an operator-valued weight εEn : A(E Prop. 12, but for our purposes here we can stay in the finite µ-index case. Corollary 16. We have dϕEn 1 + log 2π dϕˆEn
dϕˆEn 1 + log =− 2π dϕEn
Kn ≡ ni (Ln − L−n ) = −
n−1 2
log µA + log d(ρ)
n−1 2
log µA + log d(ρ) .
Proof. The von Neumann algebra A(I ) associated to an interval is a factor [8], hence, by the split property, also the von Neumann algebra A(En ) associated with the n-inter dϕˆ it ϕE
ϕˆ implement σt En on A(En ) and σ−t n val En is a factor. As both V (n) (t) and dϕEn
En
ˆ n ), we have that −2πKn is equal to log(dϕˆEn /dϕE ) plus a constant term (see on A(E n log µA + log d(ρ). Appendix C). Such a constant is fixed by Th. 15 to be n−1 2 The quantity Zn (t) ≡ (e−tKn ξn , ξn ) is the geometric partition function associated to the symmetric n-interval partition of S 1 , thus by Th. 15 Fn,µ ≡ −t −1 log Zn (t)|t=2π = − n−1 4π log µA −
1 2π
log d(ρ)
(29)
is the associated n-free energy, that we call the n-µ-free energy. The n-µ-free energy divided by the numbers of cells (intervals) gives asymptotically the mean µ-free energy. Corollary 17. The mean µ-free energy is given by 1 Fmean,µ = − 4π log µA .
Proof. Immediate by Eq. (29) we have Fmean,µ ≡ lim
1 Fn,µ n→∞ n
= =
1 log(e−2πKn ξn , ξn ) n→∞ 2πn 1 − lim n−1 log µA + 2πn log d(ρ) 4πn n→∞ 1 − 4π log µA .
= − lim
(30)
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In analogy with Sect. 3.2 the 0th and 1st spectral invariants are then defined by t log Zn (t) |t=2π , n d t log Zn (t) ≡ lim |t=2π . n→∞ dt n
a0,µ ≡ lim
(31)
a1,µ
(32)
n→∞
Note that − dtd log Zn (t) is the n − µ-energy Hn,µ associated with Zn (t). Due to the thermodynamical relation Free energy = T · Entropy − Energy, where T is the temperature, we thus define the mean n − µ-entropy by Sn,µ = t (Fn,µ + Hn,µ ) . We have: d Zn (t) d t log Zn (t) = log Zn (t) + t dt dt Zn (t) = −t Fn,µ + Hn,µ ) = −Sn,µ ,
(33) (34) (35)
thus the mean µ-entropy at the Hawking inverse temperature 2π is given by Smean,µ = lim Sn,µ /n = − lim n→∞
n→∞
d t log Zn (t) |t=2π . dt n
Proposition 18. Sn,µ = S(ϕˆEn |ϕEn ), where the latter is the Araki relative entropy between the states ϕˆEn and ϕEn . Proof. We fix a natural cone L2 (A(En ))+ (that is unique up to unitary equivalence); for example, in the vacuum representation, we can take the natural cone with respect to the vacuum vector . The derivative of log Zn (t) at t = 2π is given by d (Kn e−tKn ξn , ξn ) log(e−tKn ξn , ξn )|t=2π = − dt (e−tKn ξn , ξn ) t=2π 1/2 ξ , 1/2 ξ ) n n ρ (Kn
= −µA,n t=2π (e−tKn ξn , ξn ) = −(Kn J J 1/2 ξn , J J 1/2 ξn ) = −(Kn ξˆn , ξˆn ) ρ 1 = 2π (log ξˆn , ξˆn ) + 21 log µA,n ρ = t −1 − S(ϕˆEn |ϕEn ) + 21 log µA,n |t=2π , where ≡ ξˆ ,ξn is the Araki relative modular operator between the vectors ξn , ξˆn ∈ n ˆ n ), and J is the L2 (A(En ))+ implementing the states ϕEn on A(En ) and ϕˆE on A(E n
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213
corresponding modular conjugation. Hence d d t log(e−tKn ξn , ξn )|t=2π = log(e−tKn ξn , ξn )|t=2π + t log(e−tKn ξn , ξn )|t=2π dt dt ρ = log(e−tKn ξn , ξn )|t=2π − S(ϕˆEn |ϕEn ) + 21 log µA,n =
1 2
ρ
log µA,n − S(ϕˆEn |ϕEn ) +
1 2
ρ
log µA,n
ρ
= −S(ϕˆEn |ϕEn ) + log µA,n = −S(ϕˆEn |ϕEn ) + (n − 1) log µA + log d(ρ) which gives the thesis.
Corollary 19. We have a0,µ =
1 2
log µA ,
a1,µ = −Smean,µ = log µA − lim
1 S(ϕˆEn |ϕEn ) n→∞ n
Proof. Immediate by the above discussion.
.
By definition the µ-noncommutative Euler characteristic χA,µ is defined, in analogy with the previous sections, to be equal to 12 times the first spectral invariant. Thus we have: χA,µ ≡ 12a1,µ = −12Smean,µ . 7. CFT on a Two-Dimensional Spacetime Here we give the version of the considered asymptotic expansion in the case of a conformal QFT on a two-dimensional spacetime. The extension of the rest of our analysis is then immediate and we do not make it explicit. The model independent structure of conformal quantum field theory on the twodimensional Minkowski spacetime M2 is naturally described by a local, diffeomorphism covariant net A of von Neumann algebras A(O) associated with double cones O of M2 , see e.g. [31]. Denoting by (x, t) the space and time coordinates of a point of M2 , the restriction of A to the light axis x ± t = 0 gives rise to two local chiral conformal nets A± on R that, by conformal invariance, extend to local conformal nets on S 1 . Given the double cone O = {(x, t) : x ± t ∈ I± } associated with the intervals I+ and I− of the light axis, denote by A0 (O) the von Neumann algebra A0 (O) = A+ (I+ ) ∨ A(I− ) A+ (I+ ) ⊗ A(I− ); then A0 is a local conformal subnet of A. In the rational case one expects the subnet to have finite Jones index: [A(O) : A0 (O)] < ∞ .
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This is the case if A0 is completely rational, namely if A± are completely rational, which is automatic for example if the central charge(s) of A (i.e. of A± ) are less than one. The classification of all local conformal nets on M2 with central charge c < 1 has been obtained in [31]. We shall say that A is modular if both A+ and A− are modular. Rehren describes the structure of the inclusion A0 (O) ⊂ A(O) in terms of modular invariants [45]. The restriction to A0 of the identity representation of A decomposes as Zij ρi+ ⊗ρj− with {ρi+ } and {ρi− } irreducible sectors of A+ and A− . Accordingly, the conformal Hamiltonian H of A (the generator of the rotation one-parameter group in the time direction), has a decomposition − + e−tH = Zij e−tL0,i ⊗ e−tL0,j , i,j ± where L± 0,i is the conformal Hamiltonian of A± in the representation ρi .
Proposition 20. Let A be a modular local conformal net on the two-dimensional Minkowski spacetime. We have the expansion as t → 0+ : log Tr(e−2πtH ) ∼
2πc 1 1 2π c − log µA − t, 12 t 2 12
where c ≡ (c+ + c− )/2 is the average of the central charges c± of A± . Proof. We have the asymptotic equality as t → 0+ : − + Tr(e−2πtH ) = Zij Tr(e−2πtL0,i ) Tr(e−2πtL0,j ) i,j
∼
+
−
Zij d(ρi+ )d(ρj− ) Tr(e−2πtL0 ) Tr(e−2πtL0 )
i,j +
−
= [A : A0 ] Tr(e−2πtL0 ) Tr(e−2πtL0 ) , where we have used the Kac-Wakimoto formula in the first equality, while the identity [A : A0 ] = i,j Zij d(ρi+ )d(ρj− ) follows because i,j Zij ρi+ ⊗ ρj− is equivalent to the canonical endomorphism of A0 ⊂ A, thus + − [A : A0 ] = d Zij ρi ⊗ ρj = Zij d(ρi+ )d(ρj− ) . i,j
i,j
By [32, Prop. 24] we have the equality [A : A0 ] =
µA0 /µA .
(36)
Note that the above µ-indices are two-dimensional, while the formula in [32] concerns nets on S 1 , but the same argument entails the equality (36). Therefore we have + − 1 log Tr(e−2πtH ) ∼ log µA0 − log µA + log Tr(e−2πtL0 ) + log Tr(e−2πtL0 ) . 2
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By Prop. 3 we then obtain log Tr(e−2πtH ) πc+ 1 1 π c− 1 π c+ t ∼ log µA0 − log µA + − log µA+ − + − log µA− 2 12t 2 12 12t 2 π c− t − 12 2πc 1 1 2π ct ∼ log µA0 − log µA + − log µA0 − 2 12t 2 12 2π c 1 2πct = − log µA − , 12t 2 12 where we have made use of the identity µA0 = µA+ µA− .
In the physical context, the expansion in Prop. 20 is natural to be considered, rather than the one for the chiral components in Prop. 3. Note also that a modular net A on the two-dimensional Minkowski space is maximal if and only if log µA = 0 [31]. This is consistent with the appearance of log µA only as a first order correction to the entropy. A. Appendix. Conformal Nets on S 1 We recall here some basic facts and results about conformal nets in the form needed in the paper. We denote by I the family of proper intervals of S 1 . A net A of von Neumann algebras on S 1 is a map I ∈ I → A(I ) ⊂ B(H) from I to von Neumann algebras on a fixed Hilbert space H that satisfies: A. Isotony. If I1 ⊂ I2 belong to I, then A(I1 ) ⊂ A(I2 ). The net A is called local if it satisfies: B. Locality. If I1 , I2 ∈ I and I1 ∩ I2 = ∅ then [A(I1 ), A(I2 )] = {0}, where the brackets denote the commutator. The net A is called M¨obius covariant if it satisfies in addition the following properties C,D,E: C. M¨obius covariance. There exists a strongly continuous unitary representation U of of the M¨obius group Mob ¨ on H such that U (g)A(I )U (g)∗ = A(gI ),
g ∈ Mob, ¨ I ∈ I.
Here Mob ¨ acts on S 1 by M¨obius transformations.
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D. Positivity of the energy. The generator of the one-parameter rotation subgroup of U (conformal Hamiltonian) is positive. E. Existence of the vacuum. There exists a unit U -invariant vector ∈ H (vacuum vector), and is cyclic for the von Neumann algebra I ∈I A(I ). Let A be a M¨obius covariant net. By the Reeh-Schlieder theorem the vacuum vector is cyclic and separating for each A(I ). The Bisognano-Wichmann property then holds, see [8]: the Tomita-Takesaki modular operator I and conjugation JI associated with (A(I ), ), I ∈ I, are given by U (I (2πt)) = itI , t ∈ R,
U (rI ) = JI .
(37)
¨ of special conformal transformations Here I is the one-parameter subgroup of Mob preserving I (also called dilatations associated with I ): by identifying the upper semicircle S 1 with R ∪ {∞} via the stereographic map, thus S + with R+ , S + (t) is the map x → e−t x on R ∪ {∞}. Then I (t) is defined for any I ∈ I by conjugation by an element of Mob. ¨ U (rI ) implements a geometric action on A corresponding to the M¨obius reflection rI on S 1 mapping I onto I , i.e. fixing the boundary points of I , see [8]. Here I denotes the complement of I , I ≡ S 1 I . This immediately implies Haag duality: A(I ) = A(I ),
I ∈I,
where A(I ) is the commutant of A(I ). We shall say that a M¨obius covariant net A is irreducible if I ∈I A(I ) = B(H). Indeed A is irreducible iff is the unique U -invariant vector (up to scalar multiples), and iff the local von Neumann algebras A(I ) are factors. In this case they are III1 -factors (unless A(I ) = C identically), see [20]. Every M¨obius covariant net A decomposes uniquely into a direct integral of irreducible M¨obius covariant nets (and the analogous is true for the conformal nets below); we shall thus always assume the following. F. Irreducibility. The net A is irreducible. Let Diff(S 1 ) be the group of orientation-preserving smooth diffeomorphisms of S 1 . As is well known Diff(S 1 ) is an infinite dimensional Lie group whose Lie algebra is the Virasoro algebra. By a conformal net (or diffeomorphism covariant net) A we shall mean a M¨obius covariant net such that the following holds: G. Conformal covariance. There exists a projective unitary representation U of Diff(S 1 ) on H extending the unitary representation of Mob ¨ such that for all I ∈ I we have U (g)A(I )U (g)∗ = A(gI ), g ∈ Diff(S 1 ), U (g)xU (g)∗ = x, x ∈ A(I ), g ∈ Diff(I ), where Diff(I ) denotes the group of smooth diffeomorphisms g of S 1 such that g(t) = t for all t ∈ I . We shall say that A satisfies the split property if the von Neumann algebra A(I1 ) ∨ A(I2 ) is naturally isomorphic to A(I1 )⊗A(I2 ) when I1 and I2 are intervals with disjoint closures. The split property is entailed by the trace class condition Tr(e−tL0 ) < ∞ for all t > 0, where L0 is the conformal Hamiltonian.
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Representations. With A a local conformal net, a representation π of A on a Hilbert space H is a map I ∈ I → πI that associates to each I a normal representation of A(I ) on B(H) such that πI˜ A(I ) = πI ,
I ⊂ I˜,
I, I˜ ⊂ I .
π is said to be M¨obius (resp. diffeomorphism) covariant if there is a projective unitary representation Uπ of Mob ¨ (resp. Diff (∞) (S 1 )) on H such that πgI (U (g)xU (g)∗ ) = Uπ (g)πI (x)Uπ (g)∗ for all I ∈ I, x ∈ A(I ) and g ∈ Mob ¨ (resp. g ∈ Diff (∞) (S 1 )). Note that if π is irreducible and diffeomorphism covariant then U is indeed a projective unitary representation of Diff(S 1 ). Following [17], given an interval I and a representation π of A, there is an endomorphism of A localized in I equivalent to π; namely ρ is a representation of A on the vacuum Hilbert space H, unitarily equivalent to π, such that ρI = id A(I ) . We refer to [20] for basic facts on this structure, in particular for the definition of the dimension d(ρ), that turns out to be equal to the square root of the Jones index [34]. Let hπ be the conformal weight of the representation π , namely the lowest eigenvalue of the conformal Hamiltonian L0,π in the representation π . We shall need the following elementary fact. Lemma 21. Let A be a local M¨obius covariant conformal net on S 1 and π an irreducible representation with hπ = 0. Then π is equivalent to the identity representation. Proof. Let ξ be a unit vector such that L0,π ξ = 0. Then Uπ (g)ξ = ξ for all g ∈ Mob ¨ (see e.g. [20]). Moreover, as π is irreducible, ξ is cyclic for π . Given an interval I ∈ I and gt ≡ I (t), (t ∈ R), we have for every x ∈ A(I ), (πI (x)ξ, ξ ) = (Uπ (gt )πI (x)Uπ (gt )−1 ξ, ξ ) = (πI (U (gt )xU (gt )−1 )ξ, ξ ) . As t → ∞, U (gt )xU (gt )−1 weakly converges to (x, ), hence we have (πI (x)ξ, ξ ) = (x, ),
x ∈ A(I ) ,
yielding the statement by the uniqueness of the GNS representation.
Nets in a non-vacuum representation.. Given a conformal net A as above and a representation π of A on a Hilbert space Hπ , the map I ∈ I → Aπ (I ) ⊂ B(Hπ ) with Aπ (I ) ≡ πI (A(I )) satisfies all the above properties A to G (with Aπ and Uπ in place of A and U ), except E. We can however generalize E to E here below. A locally normal state ω on Aπ is, by definition, a family {ωI , I ∈ I}, where ωI is a normal state on Aπ (I ), such that ωI˜ Aπ (I ) = ωI
if
I ⊂ I˜ .
E . Existence of the vacuum state. There exists a locally normal state ω on Aπ that is Mob ¨ covariant: ωI = ωgI · AdUπ (g),
I ∈ I, g ∈ Mob ¨ .
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The state ω is defined by ωI ≡ (πI−1 (·), ) once we start with the vacuum representation, but E can be taken as an axiom if we start directly in the representation π . In this case, in order to obtain the vacuum representation, one can perform the GNS procedure associated with ω. One needs however to supplement E to the positivity of the energy in the vacuum state, namely ω must be a ground state. Equivalently one can require the local KMS property, that follows immediately from the above discussed Bisognano-Wichmann property if we had started from the vacuum sector. E
. Local KMS property. The modular group associated with (Aπ (I ), ωI ), I ∈ I, is AdUπ (I (−2π t)). By definition a local Mob ¨ covariant net Aπ (in a representation) is a map I ∈ I → Aπ (I ) that satisfies the properties A,B,C,D and E ,E
. We shall say that Aπ is conformal if it satisfies G and the vacuum representation is diffeomorphism covariant. Proposition 22. Let Aπ be a local Mob ¨ covariant net in a representation. There exists a local Mob ¨ covariant net A in the vacuum representation and a DHR representation π of A such that Aπ (I ) = πI (A(I )). Proof. Let {HI , σI , I } be the GNS triple associated with ωI and A(I ) ≡ σI (Aπ ). Clearly, if I ⊂ I˜, we can identify HI with a Hilbert subspace of HI˜ and I with I˜ . The usual Reeh-Schlieder analyticity argument with the KMS property E
then shows that indeed H ≡ HI = HI˜ , thus H is independent of I . The rest is now clear (cf. [21]).
B. Appendix. Trace and Determinants This appendix contains elementary known facts. Its purpose is to make explicit formula (40), as it helps to understand our definitions. Let H be an Hilbert space and ± (H) the Bose/Fermi Fock Hilbert space over H. If a ∈ B(H) and ||a|| ≤ 1 the second quantization of A± ≡ ± (a) is the linear contraction on ± (H) defined by A± ≡ 1 ⊕ a ⊕ (a ⊗ a) ⊕ (a ⊗ a ⊗ a) ⊕ · · · , ⊗ where the a ⊗ · · · ⊗ a acts on the symmetric/anti-symmetric part H± of H ⊗ · · · ⊗ H depending on the Bose/Fermi alternative. The following is well known, see e.g. [7]. n
Lemma 23. If a is selfadjoint, 0 ≤ a < 1, then Tr A± = det(1 ∓ a)∓1 , log Tr A± = ± Tr log(1 ± a).
(38) (39)
Proof. Assume first that H is one-dimensional, thus a = λ is a scalar 0 ≤ λ < 1. In ⊗n n is also one-dimensional for all n, thus we have A+ = ⊕∞ the Bose case, H+ n=0 λ , so ∞ n −1 Tr A+ = n=0 λ = (1 − λ) . For a general a (with discrete spectrum) we may decompose H = ⊕i Hi so that { } dimHi = 1 and a = ⊕i λi . Then + (H) = ⊗i i + (Hi ), where i is the vacuum vector of + (Hi ), and A+ = ⊗i + (ai ). It follows that Tr + (ai ) = (1 − λi )−1 = det(1 − a)−1 . Tr A+ = i
i
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⊗ As for the Fermi case, if H is one-dimensional then H− = {0} if n ≥ 2 and is onedimensional if n = 0, 1; if a = λ we then have A− = 1 ⊕ λ so Tr A− = 1 + λ. Since, also in the Fermi case, there is a canonical equivalence between − (a ⊕ b) and − (a) ⊗ − (b), we have Tr − (λi ) = (1 + λi ) = det(1 + a), Tr A− = n
i
i
where a = ⊕i λi . Concerning the second formula, notice that det a = eTr log a , hence log Tr A± = ∓ log det(1 ∓ a) = ∓ Tr log(1 ∓ a).
Lemma 24. Let h be a positive selfadjoint operator on H and H the Fermi second quantization of h, namely H = − (h). Then log Tr(e−tH ) = O(t) Tr(e−th )
t → 0+ .
(40)
Proof. We shall show that log 2 ≤ lim inf t→0+
log Tr(e−tH ) log Tr(e−tH ) ≤ lim sup = 1. Tr(e−th ) Tr(e−th ) t→0+
By Lemma 23 it suffices to show that log 2 ≤ lim inf t→0+
Tr log(1 + e−th ) Tr log(1 + e−th ) ≤ lim sup = 1. Tr(e−th ) Tr(e−th ) t→0+
We have log 2 · e−th ≤ log(1 + e−th ) ≤ e−th because of the corresponding function inequalities, that obviously implies the previous inequality. The Bose version of the above lemma is omitted (the U (1)-current algebra local conformal net is not rational). C. Appendix. Index and Entropy In this appendix we develop abstract mathematical results, concerning Jones index and Connes-Haagerup noncommutative measure theory, that are necessary for our work. We refer to Takesaki’s book [52] for the basic theory.
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Let R be a von Neumann algebra on a Hilbert space H, S = R its commutant. Given a n.f.s. (normal, faithful, semifinite) weight ϕ on R and a n.f.s. weight ψ on S, the dϕ is a canonical positive non-singular selfadjoint operator Connes spatial derivative dψ dϕ −it dϕ it ϕ on H such that dψ implements σt on R (the modular group of (R, ϕ)) and dψ −1 ψ dϕ implements σt on S. One has dψ = dψ . dϕ If ψ0 is another n.f.s. weight on S there holds dϕ it dϕ it = (Dψ : Dψ0 )t , (41) dψ0 dψ where (Dψ : Dψ0 ) is the unitary Connes Radon-Nikodym cocycle in S w.r.t. ψ0 and ψ. The following proposition is known. Proposition 25. Let R and S = R be von Neumann algebras on a Hilbert space H, and V a one-parameter unitary group on H such that AdV (t)R = R, t ∈ R. Given a ϕ n.f.s. weight ϕ on R such that AdV (t) R = σt , there is a unique n.f.s. ψ weight on S dϕ it such that dψ = V (t). If ψ0 is an arbitrary n.f.s. weight on S one has
where ut ≡ V (−t)
dϕ dψ0
it
(Dψ : Dψ0 )t = ut ,
(42)
.
dϕ it ϕ Proof. With ψ0 an arbitrary n.f.s. weight on S, both dψ and V (t) implements σt on 0 R, thus ut belongs to S and is a unitary σ ψ0 -cocycle. By Connes theorem, there exists a n.f.s. weight ψ on S such that ut = (Dψ : Dψ0 )t . The rest follows by formula (41).
Corollary 26. Suppose that, in Prop. 25, ϕ is the state on R given by a cyclic and separating vector ξ . If K is the infinitesimal generator of V we have ψ(1) = (e−K ξ, ξ ), 1
in particular ψ is a bounded functional iff ξ belongs to the domain of e− 2 K . Proof. Let ψ0 be the vector state on S implemented by ξ . Then ψ(1) = anal.cont. ψ0 (Dψ : Dψ0 )t t→−i
dϕ it ξ, ξ ) dψ0
= anal.cont. (V (−t) t→−i
= anal.cont. (V (−t)ξ, ξ ) = (e−K ξ, ξ ),
(43)
t→−i
where we have made use that
dϕ dψ0
is the modular operator of (R, ξ ), thus
dϕ dψ0 ξ
= ξ.
Let N1 , N2 be commuting factors on a Hilbert space H with N1 ∨ N2 = B(H). Set M1 ≡ N2 , M2 ≡ N1 , thus Ni ⊂ Mi are irreducible inclusion of factors (i = 1, 2). Let ϕi be a normal faithful state on Ni and V a one-parameter unitary group on H such that ϕ
AdV (t) N1 = σt 1 ,
ϕ
AdV (−t) N2 = σt 2 ,
t ∈ R,
where σ ϕi is the modular group of (Ni , ϕi ). Let ψ1 be the n.f.s. weight on M1 associated with V and ϕ2 by Prop. 25, namely ψ1 is characterized by
Noncommutative Spectral Invariants and Black Hole Entropy
K = log
dϕ2 dψ1
221
,
and analogously let ψ2 be the n.f.s. weight on M2 associated with V and ϕ1 . There exists a unique n.f.s. operator valued weight Ei : Mi → Ni such that ϕi ·Ei = ψi . The existence of Ei follows by the Haagerup theorem because σ ψi Ni = σ ϕi . Then Ei is faithful and unique up to a positive scalar multiple because Ni ∩ Mi = C. Proposition 27. The following are equivalent: (a) There exists a normal expectation εi : Mi → Ni ; (b) ψi is bounded. If the above hold, then Ei = ψi (1)εi and dϕ1 · ε1 + log ψ1 (1) dϕ2 dϕ2 · ε2 = log + log ψ2 (1) . dϕ1
K = − log
(44) (45)
Proof. If (a) holds, say with i = 1, then E1 = λε1 for some λ > 0, thus ψ1 = ϕ1 · E1 = ψ1 = λϕ1 · ε1 is bounded. Conversely if (b) holds then ψ1 is a normal, faithful, positive ψ linear functional on M1 whose modular group σt 1 = AdV (t) leaves N1 globally invariant, so there is a normal expectation ε : M1 → N1 by the Takesaki theorem. Clearly, if the above hold, then E1 (1) = λ, thus ψ1 (1) = ϕ1 · E1 (1) = λ, and the rest of the statement follows. Assume there exists a faithful normal expectation ε1 : M1 → N1 . Denote by ε −1 : M2 → N2 the dual operator valued weight. This is the unique n.f.s. operator valued weight M2 → N2 such that −1 dω1 · ε1 dω2 · ε −1 = dω2 dω1 for all n.f.s. weight ω1 on N1 and ω2 on N2 . According to Kosaki’s definition [33], the inclusion N1 ⊂ M1 has finite index iff ε −1 is bounded and the index is defined to be ε −1 (1), namely ε−1 = [M1 : N1 ]ε2 , where ε2 is the unique normal expectation from M2 onto N2 . Proposition 28. We have [M1 : N1 ] = ψ1 (1) · ψ2 (1) . Proof. By definition dψ1 = eK , dϕ2
dψ2 = e−K . dϕ1
Thus dϕ2 · E2 −1 dϕ1 · E1 = ; dϕ2 dϕ1
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setting λi ≡ ψi (1), since Ei = λi εi we then have dϕ2 · ε2 −1 dϕ1 · ε1 = . λ1 λ 2 dϕ2 dϕ1 On the other hand we have dϕ2 · ε2 −1 dϕ1 · ε1 = [M1 : N1 ]−1 , dϕ2 dϕ1 showing that [M1 : N1 ] = λ1 λ2 . Corollary 29. If ξi is a cyclic and separating vector for Ni such that ϕi (x) = (xξi , ξi ), x ∈ Ni , we have [M1 : N1 ] = (eK ξ1 , ξ1 )(e−K ξ2 , ξ2 ) . Suppose further that there exists a unitary U such that U M1 U ∗ = M2 , U N1 U ∗ = N2 , 1 ϕ2 = ϕ1 · AdU and U V (t)U ∗ = V (−t). Then ψ1 (1) = ψ2 (1) = [M1 : N1 ] 2 and 1
(eK ξ1 , ξ1 ) = (e−K ξ2 , ξ2 ) = [M1 : N1 ] 2 , thus 1 dϕ1 · ε1 + log[M1 : N1 ]. (46) dϕ2 2 Proof. The first equality follows by Cor. 26 and Prop. 28. The second equality then follows because U interchanges the triples of (M1 , N1 , ϕ1 ) and (M2 , N2 , ϕ2 ), thus the canonical quantities (eK ξ1 , ξ1 ) and (e−K ξ2 , ξ2 ) must coincide. The last identity (46) now follows by Eq. (45). K = − log
Araki relative entropy.. Before concluding this appendix we recall the definition of Araki relative entropy between two faithful normal states ϕ1 and ϕ2 of the von Neumann algebra M: S(ϕ1 |ϕ2 ) ≡ −(log ξ2 ,ξ1 ξ1 , ξ1 ) . Here M is in a standard form with respect to a cyclic and separating vector , the vector ξi is the canonical representative of ϕi in the natural positive cone L2 (M, )+ and ξ2 ,ξ1 is the relative modular operator, namely the polar decomposition of Sξ2 ,ξ1 is 1/2
Sξ2 ,ξ1 = J ξ2 ,ξ1 , where Sξ2 ,ξ1 is the closure of the anti-linear operator on Mξ1 defined by Sξ2 ,ξ1 xξ1 = x ∗ ξ2 . It easy to check that Sξ2 ,ξ1 = Sη2 ,η1 if η1 implements the same state of ξ1 on M and η2 implements the same state of ξ2 on M , namely ϕ1 = (·η1 , η1 )M and ψ2 ≡ ϕ2 ·AdJ = (·η2 , η2 )M . Thus S(ϕ1 |ϕ2 ) depends only on the states ϕ1 and ψ2 and we have dϕ1 S(ϕ1 |ϕ2 ) = S(ϕ1 |ψ2 ) ≡ −(log ξ1 , ξ1 ) . dψ2 We finally note, that, by taking expectation values, Eq. (46) gives dϕ1 · ε1 1 (Kξ2 , ξ2 ) = −(log ξ2 , ξ2 ) + log[M1 : N1 ] dϕ2 2 = S(ϕ2 |ϕ1 · ε1 ) + 21 H (M1 |N1 ) , where H (M1 |N1 ) = log[M1 : N1 ] is the Pimsner-Popa entropy [42].
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D. Final Comments Adding a massive charge to a black hole should increase the total mass of the black hole, hence make a change of the spacetime itself and of the entropy. In a theory of quantum gravity, the spacetime itself should be noncommutative [16] from the start. In the setting of QFT on a curved spacetime the backreaction from the gravitational field is ignored and the spacetime is classical. In the previous work [36] one considered the addition of a single charge: the increment of entropy is there a “higher order effect” and becomes visible in the associated noncommutative geometry, while the classical spacetime remains fixed [38]. The entropy in the present work also has a noncommutative geometrical nature, but rather reflects the global noncommutative geometrical complexity of the system. It would be interesting to relate our setting with Connes’ Noncommutative Geometry [15]. A link should be possible in a supersymmetric context, where cyclic cohomology appears. In this respect model analysis with our point of view, in particular in the supersymmetric frame, may be of interest. Note also that Connes’ spectral action concerns the Hamiltonian spectral density behavior, see [13].
Acknowledgement. The second named author wishes to thank, among others, I.M. Singer for an initial stimulating comment and A. Connes for a wide perspective e-mail exchange on the subject. Thanks also to B. Schroer for comments on the final manuscript.
References 1. Araki, H.: Relative Hamiltonians for faithful normal states of a von Neumann algebra. Pub. R.I.M.S., Kyoto Univ. 9, 165–209 (1973) 2. Ashtekar, A., Baez, J., Krasnov, K.: Quantum geometry of isolated horizons and black hole entropy. Adv. Theor. Math. Phys. 4, 1–94 (2001) 3. Bekenstein, J.D.: Generalized second law of thermodynamics in black hole physics. Phys. Rev. D 9, 3292–3300 (1974) 4. Bekenstein, J.D.: Holographic bound from the second low of thermodynamics. Phy. Lett. B 481, 339–345 (2000) 5. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variations. Cambridge: Cambridge Univ. Press, 1987 6. B¨ockenhauer, J., Evans, D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267–289 (2000) 7. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. Vol. 2, Texts and monographs in Physics, Berlin Heidelberg: Springer-Verlag, 1997 8. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156 201–219 (1993) 9. Buchholz, D., Wichmann, E.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986) 10. Cardy, J.L.: Operator content of two-dimensional conformally invariant theories. Nucl. Phys. B 270, 186–204 (1986) 11. Carlip, S.: Entropy from conformal field theory at Killing horizons. Class. Quantum Grav. 16, 3327– 3348 (1999) 12. Carpi, S., Weiner, M.: On the uniqueness of diffeomorphism symmetry in Conformal Field Theory. To appear in Commun. Math. Phys. DOI 10.1007/s00220-005-1335-4 (2005); Weiner, M.: Work in progress 13. Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997); Kastler, D.: Noncommutative geometry and fundamental physical interactions: The Lagrangian level – Historical sketch and description of the present situation. J. Math. Phys. 41, 3867–3891 (2000) 14. Connes, A.: On a spatial theory of von Neumann algebras. J. Funct. Anal. 35, 153–164 (1980) 15. Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994 16. Doplicher, S., Fredenhagen, K., Roberts, J.E.: Spacetime quantization induced by classical gravity. Phys. Lett. B 331(1–2), 39–44 (1994)
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17. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. I & II. Commun. Math. Phys. 23, 199–230 (1971) and 35, 49–85 (1974) 18. Evans, D.E., Kawahigashi, Y.: Quantum Symmetries on Operator Algebras. Oxford: Oxford University Press, 1998 19. Fr¨ohlich, J., Gabbiani, F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993) 20. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11 (1996) 21. Guido, D., Longo, R.: A converse Hawking-Unruh effect and dS 2 /CF T correspondence. Ann. H. Poincar´e 4(6), 1169–1218 (2003) 22. Guido, D., Longo, R., Roberts, J.E., Verch, R.: Charged sectors, spin and statistics in quantum field theory on curved spacetimes. Rev. Math. Phys. 13, 125–198 (2001) 23. Haag, R.: Local Quantum Physics. Berlin-Heidelberg-New York: Springer-Verlag, 1996 24. Haagerup, U.: Operator valued weights in von Neumann algebras. I & II. J. Funct. Anal. 32, 175–206 (1979) and 33, 339–361 (1979) 25. ‘t Hooft, G.: Dimensional reduction in quantum gravity. In: A. Aly, J. Ellis, S. Randjbar-Daemi (eds.),Salam-festschrifft, Singapore, World Scientific, 1993 26. Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. http://arxiv.org/list/math.QA/ 0406291, 2004 27. Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) 28. Kac, M.: Can you hear the shape of a drum?. Amer. Math. Monthly 73, 1–23 (1966) 29. Kac, V.G., Longo, R., Xu, F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253, 723–764 (2004) 30. Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case c < 1. Ann. of Math. 160, 493–522 (2004) 31. Kawahigashi, Y., Longo, R.: Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories. Commun. Math. Phys. 244, 63–97 (2004) 32. Kawahigashi, Y., Longo, R., M¨uger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001) 33. Kosaki, H.: Extension of Jones theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140 (1986) 34. Longo, R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126, 217– 247 (1989) 35. Longo, R.: Index of subfactors and statistics of quantum fields. II. Commun. Math. Phys. 130, 285–309 (1990) 36. Longo, R.: An analogue of the Kac-Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997) 37. Longo, R.: The Bisognano-Wichmann theorem for charged states and the conformal boundary of a black hole. Electronic J. Diff. Eq., Conf. 04, 159–164 (2000) 38. Longo, R.: Notes for a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001) 39. Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004) 40. Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998) 41. Moretti, V., Pinamonti, N.: Virasoro algebra with central charge c = 1 on the horizon of a twodimensional-Rindler space-time. J. Math. Phys. 45, 257–284 (2004) 42. Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. Ec. Norm. Sup. 19, 57–106 (1986) 43. Rehren, K.-H.: Braid group statistics and their superselection rules. In: D. Kastler (ed.), The Algebraic Theory of Superselection Sectors, Singapore, World Scientific, 1990 44. Rehren, K.-H.: Algebraic holography. Ann. H. Poincar´e 1, 607–623 (2000) 45. Rehren, K.-H.: Chiral observables and modular invariants. Commun. Math. Phys. 208, 689–712 (2000) 46. Roe, J.: Elliptic Operators, Topology and Asymptotic Methods. Pitman Res. Notes in Math. Series 395, Harlow, UK: Addison Wesley-Longman, 1998 47. Schroer, B.: Lightfront holography and the area density of entropy associated with localization on wedge regions. IJMPA 18, 1671 (2003) 48. Schroer, B., Wiesbrock, H.-W.: Modular theory and geometry. Rev. Math. Phys. 12, 139 (2000); see also: Ebrahimi-Fard, K.: Comments on: Modular theory and geometry. J. Phys. A. Math. Gen. 35(30), 6319–6328 (2000) 49. Strominger, A., Vafa, C.: Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B379, 99 (1996); Brown, J.D., Henneaux, M.: Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Commun. Math. Phys. 104, 207 (1986)
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50. Summers, S.J., Verch, R.: Modular inclusion, the Hawking temperature, and quantum field theory in curved spacetime. Lett. Math. Phys. 37, 145 (1996) 51. Susskind, L.: The world as a hologram. J. Math. Phys. 36, 6377 (1995) 52. Takesaki, M.: Theory of Operator Algebras. Vol. I, II, III, Springer Encyclopaedia of Mathematical Sciences 124 (2002), 125, 127 (2003) 53. Wakimoto, M.: Infinite Dimensional Lie Algebras. Translations of Mathematical Monographs, Vol. 195, Providence RI: Amer. Math. Soc., 2001 54. Wald, R.M.: General Relativity. Chicago, IL: University of Chicago Press, 1984 55. Xu, F.: On a conjecture of Kac-Wakimoto. Publ. RIMS, Kyoto Univ. 37, 165–190 (2001) Communicated by A. Connes
Commun. Math. Phys. 257, 227–234 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1263-8
Communications in
Mathematical Physics
A Variational Formulation for the Navier-Stokes Equation Diogo Aguiar Gomes Departamento de Matem´atica, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. E-mail: [email protected] Received: 11 May 2004 / Accepted: 1 June 2004 Published online: 22 January 2005 – © Springer-Verlag 2005
Abstract: In this paper we prove a new variational principle for the Navier-Stokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of area-preserving diffeomorphisms. This principle is a natural extension of the results by Arnold, Ebin, and Marsden concerning the Euler equation. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . 2. Navier-Stokes Equation in Magnetization Variables 3. A Variational Principle . . . . . . . . . . . . . . . 4. Diffusive Lagrangian Transformations . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The Navier-Stokes equation describes the evolution of the velocity field of a viscous incompressible fluid. Being one of the most important equations of mathematical physics, it has been studied extensively. However, its theory is still incomplete, specially in three space dimensions where general existence and uniqueness results for smooth solutions are still partial. Several authors, see for instance [Rap01, Rap00, Bus99, BFR] have studied representation formulas for solutions of the Navier-Stokes equation using probabilistic methods, and the idea of using random maps instead of deterministic ones can be traced back to Chorin [Cho73], Peskin [Pes85]. This paper is a contribuition in this direction, its main result (Theorems 1 and 2) is a new variational formulation which asserts that the solutions of the Navier-Stokes equation are critical points of a stochastic control problem on the group of area-preserving diffeomorphisms. This problem is the
Supported in part by FCT/POCTI/FEDER
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stochastic analog to the result by Arnold [Arn66], further studied by Ebin and Marsden [EM70], concerning solutions to the Euler equation. Our result is also related to the ones by Constantin in [Con01a, Con01b, Con03] as we discuss briefly in the final section of the paper. The plan of the paper is as follows: in the next section we review the magnetization formulation for the Navier-Stokes equation. Then we present a variational principle whose minimizers are solutions to the Navier-Stokes equation. Finally, we discuss the connection with the Lagrangian diffusive transformation theory developed by P. Constantin.
2. Navier-Stokes Equation in Magnetization Variables The Navier-Stokes equation in Rn for the velocity field u(x, t) of an incompressible fluid is ut + (u · ∇)u + ∇p =
1 u 2
div u = 0,
(1)
with initial condition u|t=0 = u0 . The variable p(x, t) is the pressure and is necessary to impose the incompressibility condition div u = 0. For our purposes in this paper, it is convenient to rewrite (1) in new variables, the magnetization variables. These have been used to study the Euler equation by several authors, namely Buttke [But93], Oseledets [Ose89], Russo and Smereka [RS99], among others. We will follow Chorin [Cho94] in the summary of results we present next. The magnetization variable m is obtained by adding to the velocity field u a gradient u = m + ∇k. The scalar function k(x, t) is arbitrary at t = 0 and its evolution is chosen conveniently. This transformation is a change of gauge, of which there are several possible choices, as discussed in [RS99]. Clearly, from m one can compute u by using the Leray projection on the divergence free vector fields: u = Pm. With an appropriate choice for k, the equation for the evolution of m is ∂ t m i + u j D j m i + m j D i uj =
1 mi . 2
(2)
A main difference from (1) is that Eq. (2) does not involve pressure, nor div m = 0. Furthermore, to any solution of (2) with u = Pm, corresponds a solution u to (1). In the other direction, to any solution of (1) and initial value of k there exists a solution of (2) such that u = Pm for all times.
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3. A Variational Principle This is the core section of the paper in which we prove the main results (Theorems 1 and 2). The first one asserts that any smooth solution to the Navier-Stokes equation is a critical point of variational problem. The second result states that the critical points of a closely related variational problem are solutions to the Navier-Stokes equation. Our approach is analogous to the one by Arnold [Arn66] and Ebin and Marsden [EM70]) for the Euler equation. However, the notation and methods were inspired in the paper [BCHM02]. The key idea is to replace the original variational problem on area preserving diffeomorphisms by random area preserving diffeomorphisms. Before proceeding we would like to point out that our results are formal in the sense that in all the proofs we assume some smoothness and integrability of all functions. This is partially unavoidable as there is no global regularity result to the solutions to the Navier-Stokes equation. Theorem 1. Let u be a smooth solution to the Navier-Stokes equation with smooth initial condition u0 . Define φ ω : Rn × [0, T ] → Rn by solving the random equation ∂φ ω = u ◦ Bt ◦ φ ω , ∂t
φ ω (x, 0) = x,
(3)
in which Bt is a n-dimensional Brownian motion, identified, by convenience of notation, with the shift by Bt , that is, u ◦ Bt (x, t) = u(x + Bt , t),
u ◦ Bt ◦ φ ω = u(φ ω (x, t) + Bt , t).
Similarly, let ω : Rn×n × [0, T ] → Rn×n be a fundamental solution to the equation ∂ωim + ωjm Di uj ◦ Bt ◦ φ ω = 0, ∂t
(4)
satisfying (x, 0) = I . Then (u, φ ω ) is a critical point of the functional 1 T 2 S= |u| − E ω (x, T )u0 (x), φ ω (x, T ) − x, 2 0 Rn Rn under the constraints div u = 0 and (3). Proof. Let u be a smooth solution to the Navier-Stokes equation and φ ω and ω as in the statement. Define ω : Rn × [0, T ] → Rn to be the solution to the linear equation ∂ωi + ωj Di uj ◦ Bt ◦ φ ω = 0, ∂t satisfying the initial condition ω (x, 0) = u0 (x). Given the fundamental solution ω , one can determine ω by ω (x, t) = ω (x, t)u0 (x).
(5)
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˜ φ ω ] be the augmented Lagrangian: Let S[u, T |u|2 ∂φ ω ˜ φω ] = S[u, + Eω , − u ◦ Bt ◦ φ ω n 2 ∂t R 0 −E ω (x, T )u0 (x), φ ω (x, T ) − x. Rn
Note that ω is fixed once the solution u is given, and therefore the functional S˜ does not depend on u through ω . Obviously, any critical point of S˜ which satisfies (3) is also a critical point of S under the constraint (3). Let δu be a smooth compactly supported divergence free variation of u and δφ ω a C 2 in space and C 1 in time, progressively measurable variation of φ ω . Then, using Einstein convention, T δ S˜ = uj δuj − E ωj δuj ◦ Bt ◦ φ ω Rn 0 T ∂δφjω ω ω ω + − Di uj ◦ Bt ◦ φ δφi E j ∂t Rn 0 − Eω (x, T )u0 (x), δφ ω (x, T ). Rn
Integrating by parts in time, one easily checks that T ∂δφjω ω ω ω E j − Di uj ◦ Bt ◦ φ δφi ∂t Rn 0 − Eω (x, T )u0 , δφ ω = 0. Rn
Therefore, to show that δ S˜ = 0, we must prove that T uj δuj − E ωj δuj ◦ Bt ◦ φ ω = 0. 0
Rn
(6)
Since both φ ω and Bt are measure preserving maps: ω ω ω ω −1 −1 E j δuj ◦ Bt ◦ φ = E j ◦ (φ ) ◦ Bt δuj n Rn R = E ωj ◦ (φ ω )−1 ◦ Bt−1 δuj . Rn
So we must show that
u = E ω ◦ (φ ω )−1 ◦ Bt−1 + ∇k,
since δu is divergence free. Define
m = E ω ◦ (φ ω )−1 ◦ Bt−1 ,
(7)
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and choose k such that div(m + ∇k) = 0. To prove (7) we must show that m solves (2), and so is the magnetization variable. Lemma 1. Suppose ω at t = 0 is non-random. Then 1 ∂mi + mj Di uj + uj Dj mi = mi . ∂t 2 Proof. Since ω is non-random at t = 0, the process ω ◦(φ ω )−1 ◦Bt−1 is progressively measurable. Furthermore, ω ◦ (φ ω )−1 is C 1 in time and C 2 in space. Therefore we can apply Itˆo’s formula and obtain: ∂ ∂mi = E ωi ◦ (φ ω )−1 ◦ Bt−1 ∂t ∂t ∂(φ ω )−1 ∂ωi j ω −1 −1 ω ω −1 −1 −1 =E ◦ (φ ) ◦ Bt + Dj i ◦ (φ ) ◦ Bt ◦ Bt ∂t ∂t 1 + E ωi ◦ (φ ω )−1 ◦ Bt−1 , 2 since the martingale term E Dx ωi ◦ (φ ω )−1 ◦ Bt−1 dBt vanishes. We have ω ∂i ◦ (φ ω )−1 ◦ Bt−1 = −E ωj ◦ (φ ω )−1 ◦ Bt−1 Di uj = −mj Di uj . E ∂t Since φ ω ◦ (φ ω )−1 = I d, it follows, by differentiation, −1 ∂(φ ω )−1 u ◦ Bt = −Dx (φ ω )−1 u ◦ Bt . = − Dx φ ω ◦ (φ ω )−1 ∂t Therefore
E
Dj ωi
ω −1
◦ (φ )
◦ Bt−1
∂(φ ω )−1 j ∂t
◦ Bt−1
= −E Dj ωi ◦ (φ ω )−1 ◦ Bt−1 Dk (φ ω )−1 ◦ Bt−1 uk j = −Dk E ωi ◦ (φ ω )−1 ◦ Bt−1 uk = −uj Dj mi . Finally 1 1 E ωi ◦ (φ ω )−1 ◦ Bt−1 = mi . 2 2
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Therefore we conclude that 1 ∂mi + mj Di uj + uj Dj mi = mi . ∂t 2 Since ω (x, 0) = u0 (x), the lemma implies u = m + ∇k, as required.
The second theorem in this section asserts that the critical points of the augmented Lagrangian are solutions to the Navier-Stokes equation. Theorem 2. Suppose (u, φ ω , ω ) is a smooth critical point with fixed endpoints of T |u|2 ∂φ ω Sˆ = + Eω , − u ◦ Bt ◦ φ ω , n 2 ∂t R 0 under the constraint div u = 0. Assume further that ω at t = 0 is non-random. Then u is a solution to the Navier-Stokes equation. Proof. Assume that δu, δφ ω and δω are compactly supported variations with div δu = 0, and δφ ω , δω are C 2 in space, C 1 in time, and progressively measurable. T δ Sˆ = uj δuj − E ωj δuj ◦ Bt ◦ φ ω Rn 0 T ω ∂δφ j + − Di uj ◦ Bt ◦ φ ω δφiω E ωj ∂t Rn 0 ω T ∂φj ω ω + = 0. E δj − u j ◦ Bt ◦ φ ∂t Rn 0 Thus
T
Rn
0
δωj
E
∂φjω ∂t
− u j ◦ Bt ◦ φ
ω
= 0,
which implies ∂φjω ∂t Similarly, 0
T
Rn
E
ωj
− uj ◦ Bt ◦ φ ω = 0.
∂δφjω ∂t
− D i uj ◦ B t ◦ φ
ω
δφiω
integrating by parts, we have ∂ωi + ωj Di uj ◦ Bt ◦ φ ω = 0. ∂t Finally, T uj δuj − E ωj δuj ◦ Bt ◦ φ ω = 0. 0
Rn
Since both φ ω and Bt are measure preserving maps, it follows
= 0,
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E ωj δuj ◦ Bt ◦ φ ω = E ωj ◦ (φ ω )−1 ◦ Bt−1 δuj n Rn R = E ωj ◦ (φ ω )−1 ◦ Bt−1 δuj , Rn
which implies
u = E ωj ◦ (φ ω )−1 ◦ Bt−1 + ∇k,
since u is divergence free. Define
m = E ω ◦ (φ ω )−1 ◦ Bt−1 ,
and choose k such that div(m + ∇k) = 0. We need to prove that m is the magnetization variable and solves (2). Since ω at t = 0 is non-random, Lemma 1 implies this result. 4. Diffusive Lagrangian Transformations In this section we use the formalism developed previously to give an interpretation to the theory developed by P. Constantin [Con01b] on near identity transformations to the Navier-Stokes equation. Consider a minimizer (u, φ ω ) as before and define A = E (φ ω )−1 ◦ Bt−1 . The next proposition shows that the vector A satisfies exactly the advection-diffusion equation as in [Con01b]. Proposition 1. The vector A satisfies 1 ∂A + (u · ∇)A − A = 0, ∂t 2 with A(x, 0) = x. Proof. Since (φ ω )−1 ◦ Bt−1 is non-random at t = 0 and progressively measurable, we have, proceeding as in Lemma 1, 1 ∂(φ ω )−1 ∂A ◦ Bt−1 + A =E ∂t 2 ∂t 1 = −E uDx (φ ω )−1 ◦ Bt−1 + A 2 1 = −u · ∇A + A. 2 Acknowledgements. I would like to thank Anabela Cruzeiro for pointing out a problem in an early version of this paper, as well as P. Gir˜ao for reading carefully the manuscript and bringing to my attention several important points.
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References [Arn66]
Arnold, V.: Sur la g´eom´etrie diff´erentielle des groupes de Lie de dimension infinie et ses applications a` l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16(fasc. 1), 319–361 (1966) [BCHM02] Bloch, A., Crouch, P., Holm, D., Marsden, J.: An optimal control formulation for inviscid incompressible ideal fluid flow. Proc. CDC 39, 1273–1279 (2002) [BFR] Busnello, B., Flandoli, F., Romito, M.: A probabilistic representation for the vorticity of a 3-dimensional viscous fluid and for general systems of parabolic equations. [Bus99] Busnello, B.: A probabilistic approach to the two-dimensional Navier-Stokes equations. Ann. Probab. 27(4), 1750–1780 (1999) [But93] Buttke, T.F.: Velicity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow. In: Vortex flows and related numerical methods (Grenoble, 1992), Volume 395 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Dordrecht: Kluwer Acad. Publ., 1993, pp. 39–57 [Cho73] Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57(4), 785–796 (1973) [Cho94] Chorin, A.J.: Vorticity and turbulence. Volume 103 of Applied Mathematical Sciences. New York: Springer-Verlag, 1994 [Con01a] Constantin, P.: An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14(2), 263–278 (electronic) (2001) [Con01b] Constantin, P.: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3), 663–686 (2001) [Con03] Constantin, P.: Near identity transformations for the Navier-Stokes equations. In: Handbook of mathematical fluid dynamics, Vol. II Amsterdam: North-Holland, 2003, pp. 117–141 [EM70] Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. Math. (2) 92, 102–163 (1970) [Ose89] Oseledets, V.I.: A new form of writing out the Navier-Stokes equation. Hamiltonian formalism. Uspekhi Mat. Nauk 44(3(267)), 169–170 (1989) [Pes85] Peskin, C.S. A random-walk interpretation of the incompressible Navier-Stokes equations. Commun. Pure Appl. Math. 38(6), 845–852 (1985) [Rap00] Rapoport, D.L.: Stochastic differential geometry and the random integration of the NavierStokes equations and the kinematic dynamo problem on smooth compact manifolds and Euclidean space. Hadronic J. 23(6), 637–675 (2000) [Rap01] Rapoport, D.L.: Random representations of viscous fluids and the passive magnetic fields transported on them. Discrete Contin. Dynam. Systems (Added Volume), 327–336 (2001) Dynamical systems and differential equations (Kennesaw, GA, 2000) [RS99] Russo, G., Smereka, P.: Impulse formulation of the Euler equations: general properties and numerical methods. J. Fluid Mech. 391, 189–209 (1999) Communicated by P. Constantin
Commun. Math. Phys. 257, 235–256 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1265-6
Communications in
Mathematical Physics
Generalized Complex Manifolds and Supersymmetry Ulf Lindstr¨om1,2 , Ruben Minasian3 , Alessandro Tomasiello3 , Maxim Zabzine4,5 1 2 3 4 5
Department of Theoretical Physics, Uppsala University, Box 803, 751 08 Uppsala, Sweden HIP-Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, 00014 Suomi-Finland Centre de Physique Th´eorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France LPTHE, Universit´e Pierre et Marie Curie, Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France Institut Mittag-Leffler, Aurav¨agen 17, 182 62 Djursholm, Sweden
Received: 22 May 2004 / Accepted: 19 July 2004 Published online: 11 January 2005 – © Springer-Verlag 2005
Abstract: We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two–dimensional model we construct is a supersymmetric relative of the Poisson sigma model used in the context of deformation quantization. 1. Introduction The recently developed notion of generalized complex geometry naturally extends and unifies complex and symplectic geometries, in general interpolating between the two [1–3]1 . There have been many hints that this geometry should be relevant to string theory. In this paper, we realize this expectation from a world–sheet perspective. The reasons to believe that generalized complex geometry should fit naturally in string theory basically all stem from the fact that the formalism puts the tangent T and the cotangent bundle T ∗ on the same footing, considering pairs (v, ξ ) in T ⊕ T ∗ . The basic objects of the formalism, generalized (almost) complex structures, are endomorphisms of this bundle, and admit an action not only under diffeomorphisms but also under a two–form. As we will see, this action is essentially a change in the string theory B–field. A related remark is that the structure group of this bundle is SO(d, d), which indicates a relation to the string theory T–duality group. This is strengthened by the interpolation between complex and symplectic geometry, which are mirrors in string theory. The formalism has in fact already found an application recently, from a perspective different from the one in this paper. A mirror symmetry transformation was proposed in [5] for manifolds of SU(3) structure, generalizing the case of Calabi–Yau manifolds with NS flux, considered in [6, 7]. As it turns out, mirror symmetry can be expressed naturally in terms of the T ⊕ T ∗ formalism. 1 In fact before Hitchin’s work [1] the algebraic aspects of a generalized complex (K¨ahler) geometry has been discussed in the physics literature [4].
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In [5] mirror symmetry is expressed as an exchange of two pure spinors. These objects appear in many disparate contexts, depending on which Clifford algebra one is considering. In the case of Clifford(9, 1) they play a role in Berkovits superstring [8], and in the case of Clifford(d) they can be used to define twistor spaces for manifolds of dimension d [9]. In the present paper the relevant spinors are those of the Clifford algebra naturally built on T ⊕ T ∗ , which is Clifford(d, d). The same way as usual Clifford(d) spinors can be realized in terms of (0, p) forms on an almost complex manifold, Clifford(d, d) spinors can be realized as formal sums of forms of mixed degrees. Pure spinors are then those which have a stabilizer of maximal dimension, which can be translated into an algebraic condition that we will review later. On a SU(d) structure manifold we can give two prototypical examples, which are also those exchanged by mirror symmetry: the (d/2, 0) form and an exponential of the two-form, eiJ . These pure spinors play a role in the T ⊕ T ∗ formalism: there is a correspondence between generalized complex structures and pure spinor lines. These two complementary pictures of the formalism can be seen as the two complementary pictures of string theory – from the world–sheet and from the supergravity point of view. Pure spinors naturally emerge in the low–energy context, in which, in particular, the above mentioned mirror symmetry proposal was formulated. In this paper, we are going to see how generalized complex structures emerge from the world–sheet point of view. Another aspect which is taken into account naturally by the formalism is the following. Mirror symmetry was defined in [5] in the class of manifolds of SU(3) structures. This was however a simplification. Mirror symmetry as defined in [5] is inspired by T–duality along three directions. In certain cases, essentially when the B–field has more than one leg along the T–dualized directions,2 the result of T–duality only makes sense as a “non–geometric” background. Usually, one thinks geometric quantities are sections of bundles associated with the frame bundle: they transform from chart to chart under diffeomorphisms. In string theory, the symmetry group is larger than diffeomorphisms. One can indeed use the SO(d, d) invariance mentioned above. Then, there may exist more general SO(d, d)-valued transition functions, apart from the usual Diff-valued ones. This will for example mix metric and B–field, making them not well–defined separately. In this situation one speaks of a non–geometrical background. This possibility has been emphasized in many papers; Scherk–Schwarz compactifications are for example of this type, and also the ones in [10, 11].3 Let us also emphasize that we are not assuming the existence of global isometries, and not doing T–duality. SO(d, d) only appears as a structure group. Hopefully, the structure described above will allow the formulation of mirror symmetry using pure spinors to be extended to “non–geometrical” situations. The present paper realizes SO(d, d) covariance and describes generalized complex geometry. The idea is simple and is introduced in a paper by one of the present authors [12]. The usual sigma model only contains fields in T , the images under the differential of the map X from the world–sheet to the target space. It does not contain objects in T ⊕ T ∗ . A related model is the Poisson sigma model which does contain fields both in T and T ∗ , and was used in the context of deformation quantization [13]. We mimic the structure of the Poisson sigma model for the usual one. We double the number of degrees of freedom introducing new fields η valued in the tangent T ∗ , and write an action for these 2d fields classically equivalent to the usual sigma model. 2
For simplicity, this case was not considered in [5]. Our interest in these matters owes much to a conversation with S. Hellerman, who also made the above remark about non–geometrical mirror symmetry. 3
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A difference between the two actions (the first–order one and the usual second–order sigma–model) is that while a second-order action is fully determined by the metric and a closed 3-form H , a first order action needs a section E of the O(d, d) bundle T ⊕ T ∗ . Any two such first-order actions are equivalent, i.e., lead to the same set of equations of motion, as long as they are transformed into each other by an action of a closed 2-form b. This puts the b-transform on equal footing with diffeomorphisms. This already captures some features of the formalism of generalized complex structures. More differences between the first and second order forms show up when we try to supersymmetrize the action. We are used to the idea that requiring the action to be supersymmetric constrains the target space geometry. In the second–order action this does not involve a generalized complex structure, only complex structures. In this paper we analyze the conditions under which the first order action has additional supersymmetries. This was done for N = (2, 2) supersymmetry in [12] going partly on-shell and is done here completely off-shell for N = (2, 0). Given what we already mentioned, it is fair to expect the appearance of generalized complex structures. In this paper we study models with N = (2, 0) supersymmetry (in the absence of boundaries), and find a generalized complex geometry. We consider three different cases, and the realization of the generalized complex geometry depends on the details. In particular, for a special case, at algebraic level we recover the N = (2, 2) geometry discovered in [14]4 . The first order action serves as a basis for T-duality. Since T-duality mixes the right and left sectors [16], this form of the action probes all models related by such transformations. There are many directions in which the present work might be extended. An obvious one is the inclusion of boundaries. In particular this may clarify the cases discussed in [17, 18] for which a geometrical interpretation is lacking. Further, in topological models, the relevance of generalized complex structures has been demonstrated in [19] (see also [20]). It would be interesting to twist the physical model discussed in this paper to reproduce those results in a more general setting. The structure of the paper is as follows. In Sect. 2 we introduce the first-order action and some notation. A brief review of generalized complex geometry is given in Sect. 3. We phrase the integrability conditions in local coordinates. Section 4 contains a discussion of the topological model. It represents the most general geometric situation. As the T ⊕ T ∗ formalism allows for twisting with 3-form H it is natural to examine the twisted construction in our context as well. This is done in Sect. 5 where we discuss the WZ-term. The (2, 0) sigma model is presented in Sect. 6 and the geometry of the target space is discussed. Finally, we gather the most technical part of our computations, namely the closure of the supersymmetry algebra, in an appendix.
2. First Order Actions In this section we describe the class of two dimensional models which are relevant for our discussion. We start by introducing the standard bosonic sigma model. This model has a single bosonic real field, X. X is a map from a two-dimensional world-sheet (without a boundary) to a manifold M equipped with a metric gµν and a closed three form Hµνρ . The action of the model is 4
See also [15], for recent developments.
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S=
1 2
gµν (X)dX µ ∧ ∗dX ν + Bµν (X)dX µ ∧ dX ν ,
(2.1)
where H = dB on some patch. Although B is used to write the action (2.1) down, the theory depends only on the three form H . We introduce a globally defined two-form bµν on M. Then we can define the following tensors: Eµν = gµν + bµν , E µλ Eλν = δνµ , 1 µν 1 µν Gµν = E + E νµ , θ µν = E − E νµ . 2 2
(2.2)
If g is a Riemannian metric then G is also a Riemannian metric. We can introduce a new field η which is a differential form on taking values in the pull-back by X of the cotangent bundle of M, i.e. a section of X ∗ (T ∗ M) ⊗ T ∗ . There exists a first order action [21, 12] 1 1 S= ηµ ∧ dX µ + θ µν ηµ ∧ ην + Gµν ηµ ∧ ∗ην 2 2 1 + (B − b)µν dX µ ∧ dX ν , (2.3) 2 which is equivalent to (2.1) upon the integration of η. Following the terminology proposed in [22], we call (g, b) the closed string data and (G, θ ) the open string data. We would like to stress one evident, but nevertheless important point: Despite the fact that the actions (2.1) and (2.3) are classically equivalent we need slightly different geometrical data to define them. For the second order action (2.1) we need (M, g, H ) while for the first order action (2.3) (M, g, b, H ). If db = 0 then all first order actions with different b are equivalent to the same second order action. Hence two first order actions with E µν and E˜ µν are physically equivalent if either E and E˜ are related by diffeomorphism or by a shift of the closed form (b-transform), namely Eµν − E˜ µν ∈ 2closed (M). The symmetry group relating the different (but physically equivalent) first order actions is the semidirect product of Diff (M) and 2closed (M). This observation will play an important role in further discussion5 . Another interesting property is that the action (2.3) includes the known two-dimensional topological field theories as degenerate limits. Namely if G = 0 and d(B −b) = 0 then the action (2.3) corresponds to the Poisson sigma model introduced in [23, 24], provided that θ is a Poisson tensor. In the case G = 0 and d(B − b) = 0 the model can be related to a more general type of topological theory, the WZ-Poisson sigma model [25], assuming some specific differential condition between θ and d(B −b). Presumably these topological models may arise as a result of a decoupling limit in string theory. Although the Poisson and WZ-Poisson sigma models are not the main subject of this paper, many results we present will be applicable to these models as well. 5 In this context we have a comment which is not directly relevant to the subject of this paper. Considering the properties of the first order action we could define string theory in the following fashion: Choose an open cover {Uα } of a manifold M. For each chart Uα define the first order action Sα using Eα and on the intersection Uα ∩ Uβ glue the E’s using the semidirect product of Diff (M) and 2closed (M). Now (Gα , θα ) are not tensors in the usual sense anymore since we glue them on Uα ∩ Uβ using not only Diff (M). However this “exotic” prescription does not change the physics. This remark is related to the discussion of non-geometrical string theories in [10, 11]. We hope to discuss these issues in detail elsewhere.
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The main goal of this paper is to study the extended supersymmetry of the first order action (2.3). For technical reasons related to supersymmetry it is convenient to switch to light-cone coordinates. Using (1, 0) superfields the N = (1, 0) supersymmetric version of (2.3) is S = i d 2 σ dθ D+ µ S=µ − S+µ ∂= µ − S+µ S=ν E µν + D+ µ ∂= ν (B − b)µν . (2.4) Throughout the paper we use (++, =) as worldsheet indices and (+, −) as two-dimensional spinor indices. We use (1, 0) superspace with a spinor coordinate θ . The covariant derivative D+ and supersymmetry generator Q+ satisfy 2 = i∂++ , D+
Q+ = iD+ + 2θ∂++ ,
(2.5)
where ∂++ = ∂0 ± ∂1 . In terms of the covariant derivatives, a supersymmetry transfor= mation of a superfield is given by + δm = im Q+ ,
+ δm S+ = im Q + S+ ,
+ δm S= = im Q + S= .
(2.6)
In terms of (1, 1) superfields, the N = (1, 1) first order action is given by S = d 2 σ d 2 θ D+ µ S−µ −S+µ D− µ − S+µ S−ν E µν +D+ µ D− ν (B − b)µν , (2.7) where we use the standard notation (see Appendix A in [17]). In what follows we focus on N = (1, 0) models. We would like to understand under which assumptions N = (1, 0) models admit N = (2, 0). However our results may be straightforwardly generalized to the extension of N = (1, 1) to N = (2, 1) susy. 3. Generalized Complex Geometry In this section we review some basic notions and fix notations. Namely we collect general facts concerning the generalized complex structure, see [1] and [2] for further details. Also we work out the coordinate form of the integrability conditions for the generalized complex structure. Let us start by recalling the definition of the standard complex structure on a manifold M (dim M = d). An almost complex structure is defined as a linear map on the tangent bundle J : T → T such that J 2 = −1d . This allows the definition of projectors on T , π± =
1 (1d ± iJ ). 2
(3.1)
An almost complex structure is called integrable if the projectors π± define integrable distributions on T , namely if π∓ [π± X, π± Y ] = 0
(3.2)
for any X, Y ∈ T , where [ , ] is a standard Lie bracket on T . A generalization of the notion of complex structure has been proposed by Hitchin [1]. In Hitchin’s construction T is replaced by T ⊕ T ∗ and the Lie bracket is replaced by the appropriate bracket on T ⊕ T ∗ , the so called Courant bracket. Thus a generalized
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complex structure is an almost complex structure J on T ⊕ T ∗ whose +i-eigenbundle is Courant involutive. This definition is the complex analog of a Dirac structure, a concept unifying Poisson and symplectic geometry, introduced by Courant and Weinstein [26, 27]. A detailed study of generalized complex geometry can be found in Gualtieri’s thesis [2]. Now let us give detailed definitions. On T ⊕ T ∗ there is a natural indefinite metric defined by (X + ξ, X + ξ ) = iX ξ . In the coordinate basis (∂µ , dx µ ) we can write this metric as follows 0 1d I= . (3.3) 1d 0 A generalized almost complex structure is a map J : T ⊕ T ∗ → T ⊕ T ∗ such that J 2 = −12d and that I is hermitian with respect to J , J t IJ = I. On T ⊕ T ∗ there is a Courant bracket which is defined as follows 1 [X + ξ, Y + η]c = [X, Y ] + LX η − LY ξ − d(iX η − iY ξ ). 2
(3.4)
This bracket is skew-symmetric but in general does not satisfy the Jacobi identity. However if there is a subbundle L ⊂ T ⊕ T ∗ which is involutive (closed under the Courant bracket) and isotropic with respect to I then the Courant bracket on the sections of L does satisfy the Jacobi identity. This is a reason for imposing hermiticity of I with respect to J . One important feature of the Courant bracket is that, unlike the Lie bracket, this bracket has a nontrivial automorphism defined by a closed two-form b, eb (X + ξ ) = X + ξ + iX b,
(3.5)
[eb (X + ξ ), eb (Y + η)]c = eb [X + ξ, Y + η]c .
(3.6)
such that
We can construct the projectors on T ⊕ T ∗ ± =
1 (I ± iJ ) ; 2
(3.7)
the almost generalized complex structure J is integrable if ∓ [± (X + ξ ), ± (Y + η)]c = 0,
(3.8)
for any (X + ξ ), (Y + η) ∈ T ⊕ T ∗ . This is equivalent to the single statement [X + ξ, Y + η]c − [J (X + ξ ), J (Y + η)]c + J [J (X + ξ ), Y + η]c +J [X + ξ, J (Y + η)]c = 0,
(3.9)
which resembles the definition of the Nijenhuis tensor. To relate the construction to the physical models we have to reexpress the above definitions in coordinate form. The map J can be written in the form J P J = , (3.10) LK
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where J : T M → T M, P : T ∗ M → T M, L : T M → T ∗ M and K : T ∗ M → µ T ∗ M, and hence they correspond to the tensor fields, J ν , Lµν , P µν and Kµν . Then the 2 condition J = −12d becomes µ
J µν J νλ + P µν Lνλ = −δ λ , J µν P νλ + P µν Kν λ Kµν Kν λ + Lµν P νλ Kµν Lνλ + Lµν J νλ
(3.11)
= 0, =
(3.12)
µ −δ λ ,
(3.13)
= 0.
(3.14)
The hermiticity of I with respect to J translates into the following conditions: J µν + Kµν = 0,
P µν = −P νµ ,
Lµν = −Lνµ .
(3.15)
In local coordinates the integrability condition (3.9) is equivalent to the following four conditions : µ µ ν µν ρ],ν + J ν J [λ,ρ] + P L[λρ,ν] = 0, P [µ|ν P |λρ] ,ν = 0, J µν,ρ P ρλ + P ρλ,ν J µρ − J λρ,ν P µρ + J λν,ρ P µρ λ λ λ J λν L[λρ,γ ] + Lνλ J [γ ,ρ] + J ρ Lγ ν,λ + J γ Lνρ,λ
J ν[λ J
(3.16) (3.17) − P µλ,ρ J ρν = 0,
(3.18)
+ Lλρ J γλ ,ν
(3.19)
+ J ρλ Lλγ ,ν = 0.
µ
To summarize, the generalized complex structure J is defined by three tensor fields J ν , Lµν and P µν which satisfy the algebraic conditions (3.11)-(3.15) and the differential conditions (3.16)-(3.19). The usual complex structure J is embedded in the notion of generalized complex structure J 0 . (3.20) J = 0 −J t One can check that all properties (3.11)-(3.19) are satisfied provided that J is a complex structure. Also, a symplectic structure is an example of a generalized complex structure 0 −ω−1 , (3.21) J = ω 0 where ω is an ordinary symplectic structure (dω = 0). More exotic examples exist and are given by manifolds, that do not admit any known complex or symplectic structure, but do admit a generalized complex structure [2, 28]. Consider a generalized complex structure J ; a new generalized complex structure can be generated by 1 0 1 0 J (3.22) Jb = b 1 −b 1 if b ∈ 2closed (M). The structure Jb is integrable due to the fact that the transformation (3.5) is an automorphism of the Courant bracket. The transformation (3.22) is called a b-transform and later we will see that this is related to the b-transform for the first order actions discussed in the previous section.
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The key feature of a complex manifold is that it is locally equivalent to C k via a diffeomorphism. For symplectic manifolds the Darboux theorem states that a symplectic structure is locally equivalent, via diffeomorphism, to the standard symplectic structure (R 2k , ω), where ω = dx1 ∧ dx2 + ... + dx2k−1 ∧ dx2k .
(3.23)
For generalized complex manifolds there exists a generalized Darboux theorem [2], which states that in a neighborhood of a regular point6 a generalized complex structure on a manifold M is locally equivalent via a diffeomorphism and a b-transform (see (3.22)), to the product of an open set in C k and an open set in the standard symplectic space (R d−2k , ω). The Courant bracket on T ⊕ T ∗ can be twisted by a closed three form H . Namely given a closed three form H one can define another bracket on T ⊕ T ∗ by [X + ξ, Y + η]H = [X + ξ, Y + η]c + iX iY H.
(3.24)
This bracket has similar properties to the Courant bracket. Again if a subbundle L ⊂ T ⊕ T ∗ is closed under the twisted Courant bracket and isotropic with respect to I, then the Courant bracket on the sections of L does satisfy the Jacobi identity. Thus in the integrability condition (3.9) the Courant bracket [ , ]c can be replaced by the new twisted Courant bracket [ , ]H . In local coordinates the new integrability condition is equivalent to four expressions: µ µ ν µν σ ρ],ν + J ν J [λ,ρ] + P (L[λρ,ν] + J [λ Hρ]σ ν ) = 0, P [µ|ν P |λρ] ,ν = 0, µ ρλ J ν,ρ P + P ρλ,ν J µρ − J λρ,ν P µρ + J λν,ρ P µρ − P µλ,ρ J ρν −P λσ P µρ Hσρν = 0, J λν L[λρ,γ ] + Lνλ J λ[γ ,ρ] + J λρ Lγ ν,λ + J λγ Lνρ,λ + Lλρ J λγ ,ν +Hργ ν − J λ[ρ J σγ Hν]λσ = 0.
J ν[λ J
(3.25) (3.26)
(3.27) + J λρ Lλγ ,ν (3.28)
4. Topological Model In this section we consider a toy topological model which will provide a “physical” derivation of generalized complex geometry. Also it will lead to results which will be relevant for the physical model (6.2). The model has the following action: (4.1) Stop = d 2 σ dθ S+µ ∂= µ which is part of the action (6.2). This is a topological system which describes the holomorphic maps : → M. The model is manifestly N = (1, 0) supersymmetric and can be defined over any differential manifold M. We would like to find the restrictions on M arising from the requirement that the model admits (2, 0) supersymmetry. 6 P is a Poisson structure and it will define a symplectic foliation. The point is called regular if P has constant rank in a neighborhood.
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We have to look for additional (non-manifest) supersymmetry transformations. The general transformations of S+ and are given by the following expressions: δ() µ = + D+ ν J µν − + S+ν P µν , δ()S+µ =
+
(4.2)
+
i ∂++ Lµν − D+ S+ν Kµν + + S+ν S+ρ Nµνρ + + D+ ν D+ ρ Mµνρ + + D+ ρ S+ν Qµρν . ν
(4.3)
Classically the Ansatz (4.2) and (4.3) is unique on dimensional grounds and by Lorentz covariance [12]. This Ansatz involves seven different tensors on M. We have to require the standard N = (2, 0) supersymmetry algebra, i.e. the manifest and non-manifest supersymmetry transformations commute and the nonmanifest supersymmetry transformations satisfy the following conditions [δ(2 ), δ(1 )] µ = 2i1+ 2+ ∂++ µ ,
[δ(2 ), δ(1 )]S+µ = 2i1+ 2+ ∂++ S+µ . (4.4)
Since the nonmanifest transformations are written in (1, 0) superfield then the first requirement is automatically satisfied. Next we have to calculate the commutator of two nonmanifest supersymmetry transformations. The result of the calculation is given in the Appendix. Imposing the condition (4.4) implies four algebraic and eleven differential conditions on the seven tensors introduced in (4.2) and (4.3). This fact alone shows how the problem of extended supersymmetry becomes involved when extra fields are introduced. Before analyzing the algebra in detail it is useful to look at the invariance of the action. The action (4.1) is invariant under (4.2) and (4.3) if the following algebraic conditions J µν + Kν µ = 0,
Lµν = −Lνµ ,
P µν = −P νµ ,
(4.5)
as well as the differential conditions 1 µν P ,ρ = −Nρ µν , 2
J
µ [ν,ρ]
= Qνρµ ,
1 L[µν,ρ] = Mρνµ 2
(4.6)
are satisfied. The differential conditions (4.6) allow us to express all three index tensors in terms of appropriate derivatives of two index tensors J , P , L and K. These two index tensors can be combined as a single object J P J = , (4.7) LK where J : T ⊕T ∗ → T ⊕T ∗ . It is easy to see that the algebraic part of the supersymmetry algebra (the part of (A.1,A.2) which does not involve derivatives nor three–index tensors) can be written as a single equation, namely that J 2 = −12d . Passing then to the action, the algebraic condition (4.5) is equivalent to a hermiticity of I with respect to J (i.e., the natural pairing on T ⊕ T ∗ , see the previous section). Therefore J is an almost generalized complex structure. Finally we have to analyze the eleven differential conditions coming from the algebra using (4.6). Using the results from the previous section, we see that the three differential conditions arising from (A.1) are the same as the conditions (3.16)-(3.18). The second line in (A.2) is equivalent to the condition (3.19). Surprisingly the remaining differential conditions in (A.2) are automatically satisfied provided that (3.16)-(3.19) hold and J is a almost generalized complex structure. Therefore we have proved that the differential conditions that come from the supersymmetry algebra are equivalent to integrability of J with respect to the Courant bracket.
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To summarize the topological model (4.1) admits (2, 0) supersymmetry if and only if the manifold M is a generalized complex manifold. As we briefly mentioned in the previous section, a generalized complex manifold is equivalent locally, via diffeomorphism and b-transform, to a product of a symplectic and a complex manifold. If we choose the Darboux coordinates (label n) along the sym¯ transverse to the leaf, the plectic leaf and the standard complex coordinates (label i, i) supersymmetry transformations (4.2) and (4.3) are simplified drastically and have the following form: ¯
δ i = i + D+ i , δS+i = i + D+ S+i , δ n = − + S+(n+1) , n+1
δ
+
¯
δ i = −i + D+ i , δS+i¯ = −i + D+ S+i¯ ,
δS+(n+1) = −i + ∂++ n , +
= S+n ,
δS+n = i ∂++
n+1
.
(4.8) (4.9) (4.10) (4.11)
5. Topological Model with WZ Term In the previous section we presented the topological model for which the extended supersymmetry is related to the generalized complex structure with integrability defined with the respect to the Courant bracket. The natural question is now the following: if in the integrability condition the Courant bracket is replaced by the twisted Courant bracket, can we then construct a model which incorporates twisted integrability? This is in fact possible and the solution is related to the WZ term. We consider the topological model with an additional term, 1 Stop = d 2 σ dθ S+µ ∂= µ − (5.1) d 2 σ dθ D+ µ ∂= ν Bµν . 2 The last term is a WZ term and it depends only on a closed three-form H , Hµνλ =
1 (Bµν,λ + Bλµ,ν + Bνλ,µ ), 2
(5.2)
if the world-sheet does not have a boundary. The model (5.1) has N = (1, 0) supersymmetry and can be defined over any differential manifold M equipped with a closed three-form H . The Ansatz for the nonmanifest supersymmetry transformations is given by the same expressions as before, (4.2) and (4.3). The off-shell supersymmetry algebra is exactly the same, (4.4). The main difference comes from the action. Namely invariance of the new action (5.1) under the transformations (4.2) and (4.3) leads to new relations between the three and two index tensors in the supersymmetry transformations. The action (5.1) is invariant under (4.2) and (4.3) if the following algebraic conditions are satisfied: J µν + Kν µ = 0,
Lµν = −Lνµ ,
P µν = −P νµ ,
(5.3)
as well as the differential conditions 1 µν µ P ,ρ = −Nρ µν , J [ν,ρ] + P µλ Hλνρ = Qνρµ , 2 1 1 L[µν,ρ] + J λ[µ Hν]λρ = Mρνµ . 2 2
(5.4)
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The algebraic part of all conditions remains the same as in the previous section and therefore the two-index tensors can be combined in a single object J which is an almost generalized complex structure. However the differential conditions will change. Using (5.4) we have to require that the expressions (A.1) and (A.2) reproduce the supersymmetry algebra (4.4). Using the results from Sect. 3 we see that the three differential conditions arising from (A.1) are the same as conditions (3.25)-(3.27). The second line in (A.2) is equivalent to the condition (3.28). As before the remaining differential conditions in (A.2) are automatically satisfied provided that (3.25)-(3.28) hold and that J is an almost generalized complex structure. Therefore we have proved that the differential conditions coming from the supersymmetry algebra are equivalent to integrability of J with respect to the twisted Courant bracket. 6. Sigma Model Now we turn to the “real” sigma model. For the sake of clarity, let us assume that the WZ term is absent in the action. Thus the second order N = (1, 0) action is given by S = −i d 2 σ dθ D+ µ ∂= ν Eµν ( ). (6.1) This action has the following first order form: S = i d 2 σ dθ D+ µ S=µ − S+µ ∂= µ − S+µ S=ν E µν .
(6.2)
Again, we would like to study under which conditions on the geometry of M the model (6.2) admits (2, 0) supersymmetry. We start by giving the most general Ansatz for the nonmanifest supersymmetry transµ formations. We already gave the most general Ansatz for the transformations of S+ and µ
, see (4.2) and (4.3). For S= we can write the following most general classical Ansatz for the transformations [12]: δ()S=µ = + D+ S=ν Rµν + + ∂= S+ν Zµν + + D+ ∂= ν Tµν + + S+ρ ∂= ν Uµνρ + + D+ ν S=ρ Vµν ρ + + D+ ν ∂= ρ Xµνρ + + S+ν S=ρ Yµνρ .
(6.3)
Thus altogether the supersymmetry transformations contain 14 different tensors. The commutators of non-manifest supersymmetry transformations are given in the Appendix. We have to require that (A.1) and (A.2) reduces to (4.4) (off-shell supersymmetry algebra) and that (A.3) reduces to [δ(2 ), δ(1 )]S=µ = 2i1+ 2+ ∂++ S=µ .
(6.4)
The action (6.2) is invariant under the transformations (4.2), (4.3) and (6.3) if the following algebraic conditions are satisfied: P
J µν
νµ
J νµ + Lρµ E ρν + Rµν = 0,
(6.5)
+ E Kρ + E Rρ = 0, L(νµ) + T(µν) = 0,
(6.6) (6.7)
Zρ(µ E ν)ρ − P (µν) = 0,
(6.8)
ρν
+ Tρν E
µρ
µ
µρ
ν
− Zν + Kν = 0, µ
µ
(6.9)
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as well as the following differential conditions: µ µ µ λµ + R[ν ,ρ] [ν,ρ] − V[νρ] − Mλ[νρ] E λν µ λµ − Qνρλ E νµ + (E λν Rν µ ),ρ − E λµ,ν J νρ P µλ ,ρ − E Vνρ − Yρ ρ ρ ρ ρ ρ −Uλµ − E ρν Xνλµ − Qµλ − J λ,µ + Z λ,µ − K µ,λ
J
= 0,
(6.10)
= 0,
(6.11)
= 0,
(6.12)
1 1 X[µλ]ρ + Mρ[µλ] − T[µλ],ρ + Lρ[µ,λ] + L[µλ],ρ = 0, 2 2 1 [µ ρ]ν 1 [µρ] |ρ] [µρ] [µ|ν (Zν E ), λ − E Uνλ + Nλ + P ,λ = 0, 2 2 ν|λ] −Yν [λ|ρ E |µ]ν + E νρ Nν [µλ] + E [µ|ρ = 0. ,ν P
(6.13) (6.14) (6.15)
Combining these conditions with the supersymmetry algebra we may analyze the solutions of the problem. In particular we are interested in the geometrical interpretation of the solutions. We will see that to find a general solution is hard. This is partially due to absence of appropriate mathematical notions. However we will present the solution related to the generalized complex structure as defined by Hitchin [1]. Regarding more general solutions, we can offer only some speculations, presented in Subsect. 6.2. Before turning to a discussion of possible solutions, we caution the reader that the general Ansatz we have made for the second supersymmetry will have solutions that correspond to “field equation”–type symmetries, as discussed in [12]. E.g., any transformation of the form δS+µ = + Aµν D+ F+ν ,
δS=µ = + D+ (Aνµ F=ν ) ,
(6.16)
will be a “trivial” symmetry of the (2, 0) action (6.2) if F+ν and F=ν are the S=µ and S+µ field equations, respectively.
6.1. Algebraic conditions. In this section we will analyze the content of the algebraic conditions coming from invariance of the action, (6.5–6.9), and from the algebraic part of the closure of the algebra, (A.1, A.2, A.3). For the topological model in Sect. 4, we were able to reformulate all conditions in terms of an almost complex structure J . Here we will try to get as close as we can to this doing the same for the sigma model, in particular we try to reexpress all conditions, now written in terms of d × d matrices, in terms of big 2d × 2d matrices. The reason for this is to make contact with the generalized structures. In the case at hand, the geometry can even be analyzed in terms of the usual geometric structures on the manifold (and not on T ⊕ T ∗ ), analogously to the case dubbed “generalized K¨ahler structure” in [2]. (We will find the algebraic conditions of that case as an important particular case.) We start by considering the conditions coming from the action. For example, Eqs. (6.5,6.6) can be written more elegantly as t t J L E E = − E −1 RE . (6.17) 1 1 P t Kt In what follows, we will refer to d × 2d matrices such as the one in (6.17) as “vectors”, so that the equation itself can be thought of as the vector E1 being stabilized by the matrix J t , with “eigenvalue” (−E −1 RE). If we define the projective action of GL(2d)
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B on d-dimensional matrices as CA D · E = (AE + B)(CE + D)−1 , it is easy to eliminate the eigenvalue from (6.17), to find Jt ·E = E .
(6.18)
In the old notation this equation reads J t E + Lt = E(P t E + K t ), which means that J t stabilizes E under the projective action. Turning to Eqs. (6.7, 6.8, 6.9), we put them in the form −1 E t J + IJ I = 0 , (T , −Z) J ≡J + 1 J + E −1 T P − E −1 Z . (6.19) = L+T K −Z Here I is again the metric 01 01 in (3.3). As explained there, the usual hermiticity condition for this metric reads J t IJ = I. Hence (6.19) is a hermiticity condition for J. We hasten to add that so far nothing says that this hermitian object squares to minus the identity, as was the case for J in the previous contexts. In fact, we shall see that in general this is not the case. We now move to conditions coming from closure of the algebra. Fortunately, the algebraic parts in (A.1,A.2) were already analyzed in Sect. 4. It is noticed there that they can be rewritten as the condition J 2 = −12d . Condition (A.3) is harder and requires more care. Collecting the algebraic part gives the equations RZ + ZK − T P = 0 ,
RT − ZL + T J = 0 ,
R 2 = −1 .
The first two of these read more compactly J P (T , −Z) = −R (T , −Z) . LK
(6.20)
(6.21)
Again, these conditions can be thought of as a stabilization. As for the third condition in (6.20), we will show shortly that it is implied by the other conditions we already have. (Before moving on, as a curiosity, we also notice that we can combine all of (6.20) with J 2 = −12d , to give
2 R T −Z 0 J P = −13d 0 L K
(6.22)
which thus summarizes all the algebraic equations from the algebra.) We have now rewritten all conditions in ones that involve 2d × 2d matrices. We use this to make contact with generalized structures. First of all, for the reader’s convenience we list the algebraic conditions we have found: 1. J 2= −12d(from closure of the algebra, (A.1,A.2)); 2. J t E1 = − E1 E −1 RE (6.17); Tt Tt t 3. J t −Z t = − −Z t R (6.21); −1 4. J + I JI = 0, where J ≡ J + E (T , −Z), Eq. (6.19). 1
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Having a list of objects on T ⊕ T ∗ and their conditions, it would seem natural at this point to ask to which subgroup of SO(d, d) they reduce. Unfortunately the conditions are not enough to determine a structure; there are many possible cases. This may be seen from the fact that Conditions 2 and 3 may be more or less restrictive, depending E Tt on how many columns −Z t and 1 have in common. So they can range from d to 2d independent conditions. To see this more explicitly, it is useful to change to a basis in which J simplifies. That this may be possible is again suggested by Conditions 2 and 3 above: in the extreme Tt E case in which all columns of −Z t and 1 are all independent, they can be regarded as a basis in which J is block–diagonal. Rather than doing this, we will display another Tt E change of basis, which does not rely on any assumption about the rank of −Z t 1 . The idea is to get another vector which is stabilized by J , and which cannot have any column in common with one of those we already have, E1 . The condition that this be stabilized, (6.17), implies indeed that also an orthogonal vector is stabilized: t E (1, −E)J =0. 1 This is seen to imply that (1, −E)J t = J−t (1, −E)
(6.23)
for some J− . One might hope this result, along with (6.17), can be used to produce a block–diagonalizing change of basis. However, to do that we need both right actions or both left actions. But if we transpose (6.23), we get a statement on J and not J t . A way out of this situation would be to have a hermiticity condition related to J ; we do not have −1 this, but the next best is Condition 4 above, (6.19).7 Defining X = E1 (T , −Z), this gives us −E t −E t Jt = − IJ I + X + IX t I 1 1 t 1 −E t (ZE + T ) . =− J− + 1 E −1 With this further computation, and using the action of J on the other block–vector (6.17), we obtain −1 1 1 E RE E −1 θ J t = −IE (IE)−1 , , E≡ E −E t 0 J− τ ≡ ZE t + T .
(6.24)
We have a basis in which J is block–triangular. Although we have not used Condition 3 yet, this form already shows that the stabilizer depends on the off–diagonal block τ . Rather than attempt a complete classification, we now show that the geometry can be described in terms of tensors on the manifold (which is not always the case in generalized 7 Another possibility would have been the hermitian object, J + X, also squared to minus one. Unfortunately one finds (J + X)2 = −1 + I Xt I X, a relation similar to U(d) structures on manifolds of dimension higher than d.
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complex geometry) and then return to the T ⊕ T ∗ point of view examining an important example. The geometry can be analyzed in terms of tensors on the manifold for a simple reason. It is immediate to notice that the condition J 2 = −1 implies that R 2 = −1 ,
J−2 = −1 ,
Rτ = τ J− ;
(6.25)
that is, R and J− are two almost complex structures, and τ is an intertwiner between them. (These facts could have been obtained without the change of basis; e.g., it is easy to show that Condition 2 alone is enough to assume that the “eigenvalue” R squares to minus one, and similarly from (6.23) for J− .) The fact that R squares to minus one also came more directly from the action, (of (6.20)); here we showed that it is a consequence of the other Conditions 1–4 above. This is why it was not included in that list. We still have one condition that we have not used, Condition 3 in the list above, Eq. (6.21). The condition is best analyzed after the IE change of basis. There, (6.21) reads −1 ζ ζ E RE E −1 τ = Rt ; (6.26) 0 J− − 21 g −1 τ − 21 g −1 τ ζ ≡ − 21 g −1 (T − ZE)t , and τ appears both in the matrix and in the vector, which makes the problem quadratic. Indeed, massaging the two components of these equations gives τ (J− − g −1 J−t g) = 0 ,
R(Eζ ) − (Eζ )R t =
1 −1 t τg τ . 2
(6.27)
These equations are modified (anti)–hermiticity properties on the two almost complex structures R and J− . In summary, as seen from Eqs. (6.25) and (6.27), there exist two almost complex structures, R and J− on the manifold, with an intertwiner between them, τ ; the two almost complex structures are antihermitian, one on the image and one on the kernel of this intertwiner. Notice that J− is equal to the almost complex structure of the usual sigma model (6.1) after integrating out the fields S from the first order action (6.2). 6.1.1. The hermitian case. We now analyze an example, from both the T ⊕ T ∗ and the T perspectives. Above, the problematic point was that the object which squares to minus −1 one, J , and the object which is hermitian, J + E1 (T , −Z), were not the same. To overcome this, we assume in this subsection that J t + IJ I = 0 (i.e., J ∈ O(d, d)). Our previous formulae then reduce to the “generalized K¨ahler” geometry of [2], at least as far as algebraic conditions are concerned. We start from the fact that J t stabilizes E = g + b. Under the new hermiticity assumption, this is equivalent to the following statement: [J , G] = 0 . Here G is a metric of signature d, d defined as [2] −1 −g b 1 g −1 G= = −12d + g −1 (E t 1) E g − bg −1 b bg −1
(6.28)
(6.29)
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with the property that G2 = 12d and Gt IG = I. G (or E) reduces the structure group on T ⊕ T ∗ to O(d)×O(d). J reduces the structure to U(d/2, d/2). Together, and with the compatibility condition (6.28) (or J · E = E), they reduce to U(d/2)×U(d/2). Equation (6.28) can be shown formally from the stabilization condition, but is particularly easy to see in the basis introduced above. From (6.24) and (6.29), one gets −1 1 E RE E −1 , G=E E −1 , J =E 0 −1 0 J− g I =E E −1 . (6.30) 0 −g The first equation in (6.30) is the same as Eq. (6.3) in [2],after redefining J+ ≡ E −1 RE. 1 0 1 1 (There, the change of basis E has been factorized as b 1 g −g .) As for the general case, J± are almost complex structures. However, given the hermiticity assumption, the form of the pairing I in (6.30) also shows that these two almost complex structures are both hermitian with respect to the metric g. This geometry is called (almost) bi–hermitian on the manifold. We are not done yet, because imposing hermiticity does not set T and Z to zero. What one gets is the remnant of (6.19), that is, X + IX t I. In components, this gives T t = −T and Z t = E −1 T . Using Condition 3 yields RT t − T R t = 0, hence, T is an intertwiner between R and its transpose. Equivalently, we might want to define the matrix t R 0 ˆ J = (6.31) T −R which is then an almost generalized complex structure. It is also interesting to see what happens if we slightly relax the initial condition. Looking at the triangular form for J , (6.24), a natural condition is τ = 0. (τ is only one component of X + IXt I, and thus this is weaker than the hermiticity considered above.) In this case, we have a condition similar to (6.28), namely [I J t I, G] = 0. We still have a reduction to U(d/2)×U(d/2). And we still have the two almost complex structures (we even had them in the general case). But, since we have no hermiticity, J± will no longer be hermitian with respect to the same metric g.
6.2. Differential conditions. In this subsection we discuss the differential conditions which arise both from invariance of the action and from the supersymmetry algebra. We are unable to solve the problem completely. The difficulties in finding the geneal solution may be partially ascribed to a lack of the appropriate mathematical tools. As discussed in Subsect. 6.1, even at the algebraic level the natural object J does not fit into the Hitchin framework unless extra restrictions are imposed, but setting8 Z = 0 and T = 0 (as in the solution just discussed) leads to J being an almost generalized complex structure. We consider only the case when T = Z = 0. With the differential conditions the situation is very similar. If we impose extra restrictions by hand then we may ensure that J is a generalized complex structure. E.g., ρ imposing Xµνλ = 0 and Uµν = 0 (again as in the solution above) we find that the 8 In general it is enough to put Z t = E −1 T . However for the sake of clarity we discuss only the solution T = Z = 0.
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conditions (6.12)-(6.14) coincide with the conditions (4.6), for the topological model. Therefore we can use the results from Sect. 4 and conclude that the supersymmetry algebra (A.1) and (A.2) together with (6.12)-(6.14) implies that J is a generalized complex structure. The remaining constraints that come from the invariance of the action (6.10), (6.11) and (6.15), can be rewritten as µ
V[νρ] = Lσρ E σ µ,ν + Lνσ E σ µ,ρ − Lρν,σ E σ µ , E Vνρ + Yρ λν
µ
λµ
=
Yν [λ|ρ E |µ]ν =
E νµ,ρ J λν + J λρ,ν E νµ − E λµ,ν J νρ , E µρ,ν P νλ − E λρ,ν P νµ − E νρ P µλ,ν ,
(6.32) (6.33) (6.34)
and there are eight non-trivial conditions from the algebra for S= , (A.3). From (6.32)(6.34) we derive the following differential condition for J and E: E λν E γρ (Lσρ E σ µ,ν + Lνσ E σ µ,ρ − Lρν,σ E σ µ ) + E γ µ,ν P νλ − E λµ,ν P νγ [λ γ ]ρ νµ |γ ]ρ ν −E νµ P γ λ,ν = E [γ |ρ E νµ,ρ J |λ] E − E [λ|µ Jρ, ν + J ρ,ν E ,ν E
(6.35)
which resembles a condition for the complex structure to be covariantly constant. This is indeed the interpretation for the solutions presented below. To summarize, the generalized sigma model (6.2) admits (2,0) supersymmetry (4.2), (4.3) and (6.3) (with T = Z = X = U = 0) if on M there exists a generalized complex structure J , such as (6.18), and a number of differential conditions is satisfied. Although we cannot offer an interpretation of these differential conditions in geometrical terms, it is not hard to construct additional specific examples. The main problem comes from the S= algebra. However if we assume that R is a complex structure, then there exists the coordinates when R is constant and Y = V = 0. These assumptions do solve the S= algebra (A.3), but this is not the most general solution. Using this observation we may construct various examples. We start from the simplest case with a diagonal generalized complex structure J (i.e., P = L = 0). In this case J t + R = 0 and we may use complex coordinates (the same for J and R) and assume Y = V = 0 in these coordinates. Thus the supersymmetry algebra is automatically satisfied. From (6.18) we obtain that E = J t EJ and thus E is a (1,1) tensor with respect to J . The remaining condition (6.35) implies that Ei k,j ¯ −Ej k,i ¯ = 0, which says that J is covariantly constant with respect to a connection with the torsion H = db. There exists a different way of looking for solutions. We present a solution based on two reasonable assumptions. The N = (2, 0) action (6.2) has a discrete symmetry analogous to that discussed in [12] for the N = (2, 2) model. It is invariant under S+µ → −S+µ + 2D+ λ Eλµ ,
S=µ → −S=µ − 2∂= λ Eµλ .
(6.36)
Our first assumption is that the symmetry (6.36) commutes with the second supersymmetry. This yields nine conditions on the parameter fields, four of which are P µν = 0, Nµ νρ = 0, Yµ νρ = 0, U µνρ = 0 .
(6.37)
With the additional requirement that K = −J t , we solve all conditions, algebraic as well as differential. We find that J is a complex structure which is covariantly constant µ with respect to the +-connection. I.e. writing Jµν = J ρ gρν where the hermitian metric 1 gµν ≡ 2 E(µν) , we have
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τ ∇ρ(+) Jµν = ∂ρ Jµν − (ρ[µ|τ + Hρ[µ|τ )Jν] =0,
(6.38)
where (0) is the Levi-Civita connection for gµν and the torsion is the three-form H = db. The rest of the solution is given in terms of J , E and b according to: 1 ρ L[µν] = J[ν bµ]ρ , 2 = Eµρ J ρτ E τ ν , = L[νµρ] , = Eµλ J λν,τ E τ σ + J λτ E τ σ,ν − E λσ,τ J τν ,
Lµν = Rµν 2Mµνρ Vµνσ
Qµνσ = J σ[µ,ν] .
(6.39)
In addition Tµν = 0, Zµν = 0, Xµνρ = 0 .
(6.40)
This solution may be recast in different forms using (6.38). In the first example, b ∈ 1,1 (M). Above we have analyzed the situation when b2,0 and b0,2 are allowed and the generalized complex structure has the form J 0 J = , (6.41) L −J t where L is a (2,0) and (0,2) tensor with respect to the complex structure J , such that Lij = 2bij . The metric g is hermitian with respect to J . The integrability of J implies that ∂L2,0 = 0. Analogously we consider the following generalized complex structure: J P J = (6.42) 0 −J t , where P is a (2,0) contravariant tensor with respect to the complex structure J (and J t + R = 0). P is proportional to the (2,0) part of θ (the antisymmetric part of E −1 ). Again the differential condition (6.35) can be understood as an appropriate covariantly constancy condition for J . These examples are all realized on a complex manifold M. We do not know if a generic solution is always a complex manifold. Notice that, in the first order model we have (incompletely) analyzed, there are more tensors in the game than in the second order model. Due to this, there are many more subcases that can be considered. 7. Summary In this paper our aim was to find a world-sheet realization of the generalized complex structure recently introduced by Hitchin. We have considered three different two dimensional models inspired by the first order action for the standard sigma model. The main property of these models is that the fields take values in T ⊕ T ∗ . We have found that the extended supersymmetry for these models is closely related to the generalized complex structure. This is the main result of the paper.
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We have left many unanswered questions and open problems. E.g., we were unable to find a geometrical interpretation for a generic (2,0) generalized sigma model. In general the main problem is that J does not respect the natural paring on T ⊕ T ∗ . Presumably one needs to introduce a more general Courant algebroid on T ⊕ T ∗ related to a different paring. We did not consider in detail certain models which appears naturally in the present context: the supersymmetric Poisson sigma model (i.e., when E is Poisson structure), the supersymmetric Poisson-WZ model and the generalized sigma model with WZ term. Many statements from this paper can be easily extended to these models. Finally, and maybe most unsatisfyingly, we were not able to show the possibility of having non–complex manifolds as supersymmetric backgrounds for our model (which would have been a powerful motivation for the present paper), while not being able to rule it out either. A reason for the technical complication we are facing for the “physical” first order sigma model may be related to its non–uniqueness (i. e., we could have taken other combinations of the first and second order action); maybe there are choices which make the equations simpler to solve. Independently from this, at the present level of development of the formalism, there are intrinsic technical difficulties coming for example from a big number of second and third–order tensors; presumably some better formalism to tackle them with will be needed for further progress. Acknowledgements. The work of UL is supported in part by VR grant 650-1998368. The work of RM and AT is supported in part by EU contract HPRN-CT-2000-00122 and by INTAS contracts 55-1-590 and 00-0334. We are grateful to Marco Gualtieri, Simeon Hellerman, Nigel Hitchin, Daniel Huybrechts and Pierre Vanhove for interesting discussions.
A. Appendix Through the paper we use the following conventions: A[µν] = Aµν − Aνµ ,
A(µν) = Aµν + Aνµ ,
L[µν,ρ] = Lµν,ρ + Lρµ,ν + Lνρ,µ ,
where L is antisymmetric. Below we give the complete expressions for the commutators of nonmanifest supersymmetry acting on all fields.
[δ(2 ), δ(1 )] µ = −2i1+ 2+ ∂++ λ (J µν J νλ + P µν Lνλ )
+21+ 2+ D+ S+λ (J µν P νλ + P µν Kν λ )
+21+ 2+ D+ λ D+ ρ (J νλ,ρ J µν − J
µ ν λ,ν J ρ +21+ 2+ S+λ S+ρ (P µρ,ν P νλ − P µν Nν λρ ) +21+ 2+ D+ ν S+λ (J µν,ρ P ρλ +P ρλ,ν J µρ − Qρνλ P µρ − P µλ,ρ J ρν ),
− Mνλρ P µν )
(A.1)
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[δ(2 ), δ(1 )]S+µ = −2i1+ 2+ ∂++ S+λ (Kµν Kν λ + Lµν P νλ ) +2i1+ 2+ ∂++ D+ λ (Kµν Lνλ + Lµν J νλ ) +2i1+ 2+ D+ λ ∂++ ρ (Lµν J νλ,ρ + J νλ Lµρ,ν + Kµν Lνρ,λ +2Kµν Mνρλ − 2J νρ Mµνλ − Qµλν Lνρ ) +2i1+ 2+ S+λ ∂++ ρ ×(−P νλ,ρ Lµν − Lµρ,ν P νλ + Qνρλ Kµν + 2Lνρ Nµνλ − J νρ Qµνλ ) +21+ 2+ D+ S+λ D+ ρ ×(−Kν λ,ρ Kµν −Qνρλ Kµν −Kµλ,ν J νρ + 2P νλ Mµνρ + Qµρν Kν λ ) +21+ 2+ D+ λ D+ ρ D+ γ ×(Mνλρ,γ Kµν − 2J νλ,γ Mµνρ + Mµλρ,ν J νγ + Qµρν Mνλγ ) +21+ 2+ D+ S+λ S+ρ ×(2Nν λρ Kµν + Kµλ,ν P νρ − 2Kν λ Nµνρ + P νλ Qµνρ ) +21+ 2+ S+λ S+ρ D+ γ (Nν λρ,γ Kµν + 2Qνγλ Nµνρ + Nµλρ,ν J νγ +P νλ,γ Qµνρ − Qµγν Nν λρ − Qµγλ ,ν P νρ ) +21+ 2+ D+ ρ S+λ D+ γ (Qνρλ,γ Kµν − 2Mνργ Nµνλ −2P νλ,γ Mµνρ − Mµγρ,ν P νλ − J νρ,γ Qµνλ + Qµρν Qνγλ +Qµρλ ,ν J νγ ) + 21+ 2+ S+λ S+γ S+ρ ×(2Nν λγ Nµνρ − Nµλγ,ν P νρ )
(A.2)
[δ(2 ), δ(1 )]S=µ = −21+ 2+ i∂++ S=ρ (Rν ρ Rµν ) +D+ ∂= S+ρ (Zνρ Rµν + Kνρ Zµν − P νρ Tµν ) +i∂++ ∂= ρ (Tνρ Rµν − Lνρ Zµν + J νρ Tµν ) +D+ S+ν ∂= ρ (Uσρν Rµσ + Kσν ,ρ Zµσ − P σ ν,ρ Tµσ + Kσν Uµρσ −P σ ν Xµσρ ) + S+ν D+ ∂= ρ (−Uσρν Rµσ + Qσρν Zµσ + Tµρ,σ P σ ν +P σ ν,ρ Tµσ + J σρ Uµσν + Tσρ Yµνσ ) + S+ν ∂= ρ D+ σ ×(−Uλρ ν,σ Rµλ + Qλσν ,ρ Zµλ + Xµσρ,λ P λν + P λν,ρσ Tµλ +Qλσν Uµρλ + Uµρν
λ λ ν λν λν ,λ J σ + J σ,ρ Uµλ − P ,σ Xµλρ + P ,ρ Xµσ λ +Xλσρ Yµνλ − Uλρν Vµσλ ) + i∂++ ν S=ρ ρ ×(Vλν Rµλ − Lλν Yµλρ + J σν Vµσρ ) + D+ ν D+ S=ρ ρ ρ (−Vλν Rµλ − Rµ ,λ J λν + Rσρ Vµνσ + Rσρ ,ν Rµσ ) +D+ ν D+ ρ S=σ (Vλρσ,ν Rµλ + Vµνσ ,λ J λρ − Mλνρ Yµλσ +J λρ,ν Vµλσ + Vλρσ Vµνλ ) + D+ S+ν S=ρ ×(Yσνρ Rµσ + Kσν Yµσρ − P σ ν Vµσρ ) + S+ν D+ S=ρ ρ ×(−Yσνρ Rµσ + Rµ ,λ P λν + Rσρ Yµνσ ) + S+ν S=ρ D+ σ
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νρ ρ νρ λ λν + Yµ ,λ J λσ + Qλσν Yµλρ ,σ Rµ + Vµσ ,λ P ρ ρ νρ λ ν ρ +Vλσ Yµνλ + P λν ,σ Vµλ − Yλ Vµσ ) + i∂++ ∂= ×(−Lσ ν,ρ Zµσ + J σν,ρ Tµσ − Lσ ν Uµρσ + J σν Xµσρ + Xσ νρ Rµσ ) +∂= S+ν S+ρ (−2Nσ[νρ] Lµσ − Zµν ,σ P σρ + P σ ν Uµσρ − Zσν Yµρσ ) νρ ρ νρ +S+ν S+ρ ∂= σ (−Nλ ,σ Zµλ + Uµσ ,λ P λν − Nλ Uµσλ ρ −P λρ,σ Uµλν + Uλσ Yµνλ ) + D+ ∂= ν D+ ρ
×(−Yλ
×(−2Mσ [νρ] Zµσ + Tµν,λ J λρ + 2J λ[ρ,ν] Tµλ − J σν Xµρσ +Xσρν Rµσ − Tσ ν Vµρσ − Tσ ν,ρ Rµσ ) + D+ ν D+ ρ ∂= σ ×(−Mλνρ,σ Zµλ − Xµρσ,λ J λν − J λν,ρσ Tµλ − Mλνρ Uµσλ −J λν,ρ Xµλσ + J λρ,σ Xµνλ + Xλρσ,ν Rµλ + Xλρσ Vµνλ ) ρ
ρ
−D+ ν ∂= S+ρ (Qλν Zµλ + Zµ
λ ,λ J ν
+ P λρ,ν Tµλ + P λρ Xµνλ ρσ λν ,λ P
−Zσρ Vµνσ − Zσρ ,ν Rµσ ) + S+ν S+ρ S=σ (Yµ νρ ρσ −Nλ Yµλσ + Yλ Yµνλ ) .
(A.3)
References 1. Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54(3), 281–308 (2003) 2. Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis. http://xxx.lanl.gov/abs/math.DG/0401221, 2004 3. Huybrechts, D.: Generalized Calabi-Yau structures, K3 surfaces, and B-fields. http://arxiv.org/abs/math.AG/0306162, 2003 4. Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: The case of complex tori. Commun. Math. Phys. 233, 79 (2003) 5. Fidanza, S., Minasian, R., Tomasiello, A.: Mirror symmetric SU(3)-structure manifolds with NS fluxes. http://arxiv.org/abs/hep-th/0311122, 2003 6. Vafa, C.: Superstrings and topological strings at large N. J. Math. Phys. 42, 2798 (2001) 7. Gurrieri, S., Louis, J., Micu, A., Waldram, D.: Mirror symmetry in generalized Calabi-Yau compactifications. Nucl. Phys. B 654, 61 (2003) 8. Berkovits, N.: Super-Poincare covariant quantization of the superstring. JHEP 0004, 018 (2000) 9. Lawson, H.B., Michelsohn, M.L.: Spin Geometry. Princeton, NJ: Princeton Univ. Press, 1989 10. Hellerman, S., McGreevy, J., Williams, B.: Geometric constructions of nongeometric string theories. JHEP 0401, 024 (2004) 11. Flournoy, A., Wecht, B., Williams, B.: Constructing nongeometric vacua in string theory. http://arxiv.org/abs/hep-th/0404217, 2004 12. Lindstrom, U.: Generalized N = (2,2) supersymmetric non-linear sigma models. Phys. Lett. B 587, 216–224 (2004) 13. Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000) 14. Gates, S.J., Hull, C.M., Rocek, M.: Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models. Nucl. Phys. B 248, 157 (1984) 15. Lyakhovich, S., Zabzine, M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B 548, 243 (2002) 16. Hassan, S.F.: O(D,D:R) Deformations of Complex Structures And Extended World Sheet Supersymmetry. Nucl. Phys. B 454, 86 (1995) 17. Lindstrom, U., Zabzine, M.: N = 2 boundary conditions for non-linear sigma models and LandauGinzburg models. JHEP 0302, 006 (2003) 18. Lindstrom, U., Zabzine, M.: D-branes in N = 2 WZW models. Phys. Lett. B 560, 108 (2003) 19. Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49–81 (2004)
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20. Grange, P.: Branes as stable holomorphic line bundles on the non-commutative torus. JHEP 0410, 002 (2004) 21. Baulieu, L., Losev, A.S., Nekrasov, N.A.: Target space symmetries in topological theories. I. JHEP 0202, 021 (2002) 22. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999) 23. Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235, 435 (1994) 24. Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A 9, 3129 (1994) 25. Klimcik, C., Strobl, T.: WZW-Poisson manifolds. J. Geom. Phys. 43, 341 (2002) 26. Courant, T.: Dirac manifolds. Trans. Amer. Math. Soc. 319(2), 631–661 (1990) 27. Courant, T., Weinstein, A. Beyond Poisson structures. In: Action hamiltoniennes de groupes. Troisi`eme th´eor`eme de Lie (Lyon, 1986), Travaux en Cours, 27, Paris: Hermann, 1988, pp. 39–49 28. Cavalcanti, G., Gualtieri, M.: Generalized complex structures on nilmanifolds. http://arxiv.org/abs/math.DG/0404451, 2004 Communicated by M.R. Douglas
Commun. Math. Phys. 257, 257–272 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1268-3
Communications in
Mathematical Physics
Strict Quantizations of Almost Poisson Manifolds Hanfeng Li Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada. E-mail: [email protected] Received: 20 May 2003 / Accepted: 28 September 2004 Published online: 11 January 2005 – © Springer-Verlag 2005
Abstract: We show the existence of (non-Hermitian) strict quantization for every almost Poisson manifold. 1. Introduction In the passage from classical mechanics to quantum mechanics, smooth functions on symplectic manifolds (more generally, Poisson manifolds) are replaced by operators on Hilbert spaces, and the Poisson bracket of smooth functions are replaced by commutators of operators. When one thinks of classical mechanics as limits of quantum mechanics, the Poisson bracket becomes limits of commutators. Based on the general theory of formal deformations of algebras [9], F. Bayer et al. [1] initiated the study of deformation quantization of Poisson manifolds. Let M be a Poisson manifold. Denote by C ∞ (M) the space of smooth C-valued functions on M, and denote by C ∞ (M)[[]] the space of formal power series with coefficients in C ∞ (M). Recall that a star product on M is a C[[]]-bilinear associative multiplication ∗ on C ∞ (M)[[]] of the form f ∗g =
∞
Cr (f, g)r ,
for f, g ∈ C ∞ (M),
r=0
where C0 (f, g) = f g, f ∗ g − g ∗ f ≡ {f, g}i mod 2 , and each Cr (·, ·) is a bidifferential operator. The algebra (C ∞ (M)[[]], ∗) is called a deformation quantization of M. The existence of deformation quantizations for any symplectic manifold was proven first by De Wilde and Lecomte [7]. The general case of Poisson manifolds was proven by Kontsevich [12]. In deformation quantizations is only a formal parameter, and elements in C ∞ (M)[[]] are not operators on Hilbert spaces. In order to study quantizations in a stricter sense, Rieffel introduced strict deformation quantization of Poisson manifolds
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[23, 24], and showed that noncommutative tori arise naturally as strict deformation quantizations of certain Poisson brackets on the ordinary torus. Later, Landsman introduced the weaker notion of strict quantization to accommodate some other interesting examples such as Berezin-Toeplitz quantization of Kahler ¨ manifolds. Recall the definition of strict quantization as formulated in [27, 13]: Definition 1.1. Let M be a Poisson manifold, and let C∞ (M) be the algebra of continuous functions on M vanishing at ∞. By a strict quantization of M we mean a dense ∗-subalgebra A of C∞ (M) closed under the Poisson bracket, together with a continuous field of C ∗ -algebras A over a closed subset I of the real line containing 0 as a non-isolated point, and linear maps π : A → A for each ∈ I , such that (1) A0 = C∞ (M) and π0 is the canonical inclusion of A into C∞ (M), (2) the section (π (f )) is continuous for every f ∈ A, (3) for all f, g ∈ A we have lim [π (f ), π (g)]/(i) − π ({f, g}) = 0.
→0
If each π is injective, we say that the strict quantization is faithful. If A ⊇ Cc∞ (M), the space of compactly supported smooth functions on M, we also say that the strict quantization is flabby. If (π (f ))∗ = π (f ∗ ) for all ∈ I and f ∈ A, we say that the strict quantization is Hermitian. When a Lie group G has a smooth action α on M preserving the Poisson bracket, if G also has a strongly continuous action β on each A such that β0 = α ∗ and the maps π are all G-equivariant, we say that the strict quantization is G-equivariant. When the strict quantization is faithful and π (A) is a ∗-subalgebra of A for each , it’s called a strict deformation quantization. Strict quantizations have been constructed for several classes of Poisson manifolds such as Poisson manifolds coming from actions of Rd [26], quantizable compact Kahler ¨ manifolds [2], dual of integrable Lie algebroids [25, 15], compact Riemannian surfaces of genus ≥ 2 [11, 17, 18], etc. These constructions are all global, and the resulting strict quantizations are Hermitian. However, the progress of the study of strict quantizations is much slower compared with that of deformation quantizations–so far there is even no existence result for general symplectic manifolds. Recently Natsume et al. [19] constructed strict quantizations for every compact symplectic manifold M satisfying the topological conditions that π1 (M) is exact and π2 (M) = 0. Roughly speaking, they use partition of unity to reduce M to Darboux charts, where they can use the Moyal-Weyl product. Thus their construction is local. It turns out that the resulting strict quantizations are not Hermitian. Recall that an almost Poisson manifold is a smooth manifold M equipped with some ∈ (∧2 T M) [6]. In this case, we can still define a bracket {f, g} = (df, dg) for f, g ∈ C ∞ (M), which is bilinear and skew-symmetric, and satisfies the Leibniz rule. And the bracket satisfies the Jacobi identity if and only if M equipped with this bracket is actually a Poisson manifold. Clearly we can also talk about strict quantizations of almost Poisson manifolds. The main result of this paper is the following: Theorem 1.2. Let (M, ) be an almost Poisson manifold, and let α be a smooth action of a Lie group G on M preserving the bracket. If M has a G-invariant Riemannian metric, then M has a G-equivariant faithful flabby strict quantization over I = [0, 1] with A = Cc∞ (M). In particular, taking G = {e} we see that M has a faithful flabby strict quantization over [0, 1] with A = Cc∞ (M).
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Our construction is also local, but different from the one in [19]. Actually we shall construct a locally trivial C ∗ -algebra bundle over M in a canonical way, thus don’t need local charts and partition of unity. But our strict quantizations are not Hermitian either. This paper is organized as follows. Though our construction for strict quantizations of almost Poisson manifolds is only slightly more complicated for that of symplectic manifolds, the idea is most natural in the case of symplectic manifolds. So we prove Theorem 1.2 for symplectic manifolds first in Sect. 2. Then we prove Theorem 1.2 for the general case in Sect. 3. Our construction depends on the choice of an inner product on the vector bundle T ∗ M ⊕T M. We define homotopy of strict quantizations in Sect. 4, and show that the homotopy class of our strict quantizations doesn’t depend on the choice of the inner products. In Sect. 5 we define local strict quantizations, and show that they can’t be Hermitian. We also show that our strict quantizations can’t be restricted to a ∗-subalgebra of A to get a strict deformation quantization of M unless = 0. This gives a negative answer to a question of Rieffel [27, Question 25]. In Sect. 6 we discuss certain functorial properties of our construction. All of our construction is based on the existence of asymptotic representations of Heisenberg commutation relations (Definition 2.1). We prove the existence of such asymptotic representations in Sect. 7. 2. Strict Quantizations of Symplectic Manifolds Throughout this paper, for a continuous field D of C ∗ -algebras {D,x }x∈X over a locally compact Hausdorff space X we denote by ∞ (D) the algebra of continuous sections of D vanishing at ∞ [8]. We show the main idea of our construction first. Let (M, ω) be a symplectic manifold, and let ∈ (∧2 T M) be the corresponding bivector field as usual. Let f, g ∈ Cc∞ (M). Since we think of A as deformations of C∞ (M), we would like to write π (f ) as f + τ (f ), which makes sense when A contains C∞ (M) as a C ∗ -subalgebra, and assume that τ (f ) → 0
(1)
as → 0. Assume further that C∞ (M) lies in the center of A . Then [π (f ), π (g)]/(i) − π ({f, g}) = [τ (f ), τ (g)]/(i) − {f, g} − τ ({f, g}). Thus the condition (3) in Definition 1.1 becomes [τ (f ), τ (g)]/(i) − {f, g} → 0. Notice that {f, g} doesn’t depend on f and g, but depends only on df and dg. So we would like to assume that τ (f ) depends only on df linearly. Then we attempt 1 to write τ (f ) as 2 ϕ (df ), where ϕ : (T ∗ M) → A is a linear map. Now [τ (f ), τ (g)]/(i) − {f, g} → 0 becomes [ϕ (df ), ϕ (dg)] − (df, dg)i → 0.
(2)
We assume further that A = ∞ (D) for some continuous field of C ∗ -algebras {D,x }x∈M over M, and that D,x contains T ∗ Mx as a linear subspace with ϕ being just pointwise embedding. Then (2) becomes [u, v] − x (u, v)i D,x → 0 for all u, v ∈ T ∗ Mx .
(3)
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This leads to our definition of asymptotic representation of Heisenberg commutation relations and that of Heisenberg C ∗ -algebra A2n (which will be D,x for 2n = dim M) in Definition 2.1 below. In order to embed T ∗ Mx into A2n without referring to local basis, we also need an action of the structure group of T ∗ M on A2n . If we consider only , then the structure group is the symplectic linear group Sp(2n) [5], which is too big. By adding a compatible almost complex structure on T ∗ M we can reduce the structure group to the unitary group U (n) (see Lemma 2.5). Here we recall how U (n) acts on T ∗ Mx . Let V be a finite dimensional vector space over R with a symplectic structure ω and a compatible almost complex structure J [5], i.e. J : V → V is linear satisfying that J 2 = −1 and < u, v >:= ω(u, J v) is an inner product on V . Say dimV = 2n. Then we can always find basis , v1 , · · · , vn of V such that under this basis u1 , · ·· , un 0 I 0 −I ω and J have matrix forms and respectively. We’ll call such basis −I 0 I 0 a unitary basis of V . Notice that a unitary basis is an orthonormal basis under the induced inner product. If we make V into a complex vectorspace by J and identify X −Y matrix X + iY ∈ Mn×n (C), where X, Y ∈ Mn×n (R), with ∈ M2n×2n (R), Y X then U (n) is exactly the group of linear transformations on V taking unitary bases to unitary bases. Definition 2.1. Let n ∈ N, and let R2n be equipped with the standard symplectic vector space structure and the standard compatible almost complex structure, i.e. for the standard basis e1 , · · · , e2n being a unitary basis. By an asymptotic representation of Heisenberg commutation relations, we mean a unital C ∗ -algebra A2n with a strongly continuous action ρ of U (n) and a U (n)-equivariant R-linear map ϕ : R2n → A2n for each 0 < ≤ 1 such that (1) for any u, v ∈ R2n we have [ϕ (u), ϕ (v)] → (u, v)i as → 0; (2) the map (0, 1] → B(R2n , A2n ) given by → ϕ is continuous, where B(R2n , A2n ) is the Banach space of linear maps R2n → A2n ; 1 (3) 2 ϕ → 0 as → 0; (4) A2n is generated by ∪0<≤1 ϕ (R2n ). The C ∗ -algebra A2n will be called a Heisenberg C ∗ -algebra of dimension 2n. Remark 2.2. The condition (3) is not crucial. Given A2n and ϕ satisfying the other conditions we can always reparameterize ϕ ’s to make them satisfy (3). The main technical part of our construction is the following: Theorem 2.3. For each n ∈ N there exists a Heisenberg C ∗ -algebra A2n . Theorem 2.3 will be proved in Sect. 7. From now on we’ll fix a Heisenberg C ∗ -algebra A2n for each n ∈ N unless stated otherwise. We’ll construct a C ∗ -algebra bundle D over M with fibres all isomorphic to A2n and a bundle map T ∗ M → D. Actually we shall do this construction more generally for almost symplectic bundles, which will be useful in Sect. 3. Definition 2.4. Let E → M be a real vector bundle, and let ∈ (∧2 E ∗ ). We call the pair (E, ) an almost Poisson bundle over M. If is nondegenerate everywhere we call it an almost symplectic bundle over M. For two almost Poisson bundles (E, E ) and (F, F ) over M we call a bundle map ψ : E → F an almost Poisson map if F (ψ(f ), ψ(g)) = E (f, g) for all f, g ∈ (E).
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As the case for symplectic manifolds [5], one sees easily that every almost symplectic bundle (E, ) has compatible almost complex structures. Lemma 2.5. Let (E, ) be an almost symplectic bundle over M with a compatible almost complex structure J . Say dimE = 2n. Then for any x ∈ M there exists a neighborhood U of x and sections f1 , · · · , fn , g1 , · · · , gn of E on U such that they are a unitary basis at each point of U . Proof. It is easy to see that we can find sections f1 , · · · ,fn , g1 , · · · , gn near x such 0 I that under this basis, is of the matrix form at each point. Let E + = −I 0 span{f1 , · · · , fn }. Then near x this is a subbundle of E, and Ey+ is a Lagrangian subspace of Ey at each point y. Let f1 , · · · , fn be n sections of E + near x such that f1 (y), · · · , fn (y) is an orthonormal basis at each point y. Then f1 , · · · , fn , Jf1 , · · · , Jfn satisfy the requirement. Lemma 2.5 shows that the structure group of (E, , J ) is U (n). Let U (E) be the set of unitary bases of E at all points, then U (E) is a principle U (n)-bundle on M. As usual, the action ρ of U (n) on A2n induces a C ∗ -algebra bundle D = U (E) ×U (n) A2n over M, which is the quotient of U (E) × A2n by the relation (a, T ) ∼ (ag, g −1 T ) with g ∈ U (n). Then D has all fibres isomorphic to A2n . Notice that the induced vector bundle E := U (E) ×U (n) R2n has an induced almost symplectic structure and an induced almost complex structure. Clearly E equipped with these structures is isomorphic to (E, , J ). For each ∈ (0, 1] the U (n)-equivariant linear map ϕ : R2n → A2n also induces a bundle map E ∼ = E → D, which we still denote by ϕ . Definition 2.6. Let (E, , J ) be as in Lemma 2.5. We call the C ∗ -algebra bundle D constructed above the quantization bundle of (E, , J ), and call the bundle maps ϕ : E → D the quantization maps. Conventions 2.7. Let E and D be a real vector bundle and a C ∗ -algebra bundle over M respectively. Every R-linear bundle map ϕ : E → D extends to a C-linear bundle map from the complexified bundle E ⊗ C to D sending f + ig to ϕ(f ) + iϕ(g), where f, g ∈ (E). We’ll denote this extended map still by ϕ. Since A2n is unital the bundle D contains the trivial bundle M × C as a subbundle naturally. Thus ∞ (D) contains C∞ (M) as a subalgebra. We are ready to construct strict quantizations for M: Theorem 2.8. Let (M, ) be a symplectic manifold. Let J be a compatible almost complex structure on T ∗ M, and let D and ϕ be the quantization bundle and maps of 1 (T ∗ M, , J ). Let A = ∞ (D) and π (f ) = f + 2 ϕ (df ) for all 0 < ≤ 1 ∞ and f ∈ Cc (M). Also let A0 = C∞ (M), and let π0 be the canonical embedding of Cc∞ (M) into C∞ (M). Let {A } be the subfield of the trivial continuous field of C ∗ -algebras [0, 1] × ∞ (D) over [0, 1]. Then {A , π } is a faithful flabby strict quantization of M over I = [0, 1] with A = Cc∞ (M). If a Lie group G has a smooth action on M preserving and J , then this strict quantization is G-equivariant. Proof. We verify the conditions in Definition 1.1. Condition (1) follows from our choice of (A0 , π0 ). Condition (2) follows from Definition 2.1(2),(3). Condition (3) follows from Definition 2.1(1) and our discussion at the beginning of this section. Thus {A , π } is a
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strict quantization of M. Since A = Cc∞ (M) this strict quantization is flabby. The faithfulness follows from Lemma 2.9 below. In fact we need only ϕ (R2n ⊗C)∩C1A2n = {0} here, but we shall need the full result of Lemma 2.9 later in Corollary 5.6. Lemma 2.9. Let A2n be a Heisenberg algebra. Let V = ϕ (R2n ⊗C)+(ϕ (R2n ⊗C))∗ . Then V ∩ C1A2n = {0} for every 0 < ≤ 1. Proof. It suffices to show that the only U (n)-fixed element in V is 0. Endow U (n) with U (n) defined the normalized Haar measure. Let σ be the canonical map A2n → (A2n ) by σ (a) = U (n) ρh (a) dh. Similarly define τ : R2n ⊗ C → (R2n ⊗ C)U (n) . Then ϕ ◦ τ = σ ◦ ϕ . Clearly (R2n )U (n) = {0}, and hence τ (R2n ⊗ C) = {0}. Consequently (V )U (n) = σ (V ) = {0}. Finally, the assertions about the G-action are clear.
The proof of Theorem 2.8 generalizes to Theorem 2.10 immediately: Theorem 2.10. Let (M, ) be an almost Poisson manifold. Let (E, E , JE ) be as in Lemma 2.5. Suppose that ψ : T ∗ M → E is an almost Poisson map (Definition 2.4). Let D and ϕ be the quantization bundle and maps of (E, E , JE ). Let A = ∞ (D) and 1 π (f ) = f + 2 (ϕ ◦ ψ)(df ) for all 0 < ≤ 1 and f ∈ Cc∞ (M). Also let A0 and ϕ0 be as in Theorem 2.8. Let {A } be the subfield of the trivial continuous field of C ∗ -algebras [0, 1] × ∞ (D) over [0, 1]. Then {A , π } is a faithful flabby strict quantization of M over I = [0, 1] with A = Cc∞ (M). If a Lie group G has a smooth action on M preserving and has a smooth action on (E, E , JE ) such that the projection E → M and the map ψ are G-equivariant, then this strict quantization is G-equivariant. 3. Strict Quantizations of Almost Poisson Manifolds Throughout the rest of this paper (M, ) will be an almost Poisson manifold unless stated otherwise. To construct strict quantizations of M, by Theorem 2.10 it suffices to find (E, E ) and ψ : T ∗ M → E as in Theorem 2.10. In fact there is a canonical choice of such (E, E ) and ψ. This follows from Lemmas 3.1 and 3.2, for which we omit the proof. Lemma 3.1. Let E → M be a vector bundle, and let E ∗ be its dual bundle. Then x ((u1 , v1∗ ), (u2 , v2∗ )) =< u1 , v2∗ > − < u2 , v1∗ > defines an almost symplectic structure on E ⊕ E ∗ , where uj ∈ Ex , vj∗ ∈ Ex∗ and < ·, · > is the canonical pairing between E and E ∗ . Lemma 3.2. Let (E, E ) be an almost Poisson bundle over M, and let σ : E → E ∗ be the induced bundle map defined by σx (u) = E,x (·, u). Let E ⊕ E ∗ be endowed with the canonical almost symplectic structure defined in Lemma 3.1. Then the bundle map ψ : E → E ⊕ E ∗ defined by ψx (u) = ( √1 u, √1 σx (u)) is an almost Poisson map. 2
2
By Theorem 2.10 we have: Theorem 3.3. Let M be the canonical almost symplectic structure on T ∗ M ⊕ T M defined in Lemma 3.1 for E = T ∗ M. Let ψ : T ∗ M → T ∗ M ⊕ T M be the bundle map defined in Lemma 3.2 for (T ∗ M, ). Let J be a compatible almost complex structure for (T ∗ M ⊕ T M, M ), and let D and ϕ be the quantization bundle and maps of (T ∗ M ⊕ T M, M , J ). Let {A , π } be as in Theorem 2.10. Then {A , π } is a faithful flabby strict quantization of M over I = [0, 1] with A = Cc∞ (M). If a Lie group G has a smooth action on M preserving and J , then this strict quantization is G-equivariant.
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We are ready to prove Theorem 1.2: Proof of Theorem 1.2. Given any Riemannian metric on M, the induced isomorphism T M → T ∗ M induces an inner product on T ∗ M, and hence induces an inner product on T ∗ M ⊕ T M by requiring T ∗ M and T M to be perpendicular to each other. Therefore T ∗ M ⊕ T M has a G-invariant inner product. Notice that given an inner product on an almost symplectic bundle (E, ) there is a canonical way to construct a compatible almost complex structure J on E [5]. Thus T ∗ M ⊕ T M has a G-invariant compatible almost complex structure. Now the assertions follow from Theorem 3.3. Corollary 3.4. For any smooth action of a compact Lie group G on M preserving , there is a G-equivariant faithful strict quantization of M over I = [0, 1] with A = Cc∞ (M). Proof. For any smooth action of a compact Lie group, the manifold admits an invariant Riemannian metric by integrating any given Riemannian metric. Rieffel showed [23] that there is no strict deformation quantization of the rotationally invariant symplectic structure on S 2 respecting the action of SO(3). So this gives us some sign on how much more restrictive strict deformation quantizations are than strict quantizations. Corollary 3.5. If a Lie group G has a bi-invariant Riemannian metric, then g∗ equipped with the Lie-Poisson bracket [6] admits a faithful strict quantization with A = Cc∞ (g∗ ) equivariant under the coadjoint action of G, where g is the Lie algebra of G and g∗ is the dual. Proof. Identify the cotangent space of g∗ at each point with g. Then for any ξ ∈ g∗ ∗ and g ∈ G, the isomorphism Tξ∗ g∗ → TAd g∗ is exactly Adg : g → g. In particular ∗ g (ξ ) considering ξ = 0, we see that g∗ admits an invariant Riemannian metric if and only if the vector space g has an inner product invariant under the adjoint action of G, if and only if G has a bi-invariant Riemannian metric. Example 3.6. Assume that T M is trivial. Let X1 , · · · , Xm ∈ (T M), giving the trivial∗ ∈ (T ∗ M) be the dual basis. Define J on T ∗ M ⊕ T M ization of T M. Let X1∗ , · · · , Xm ∗ by J (Xk ) = Xk and J (Xk ) = −Xk∗ for 1 ≤ k ≤ m. Then the quantization bundle D is the trivial bundle M × A2m , and thus ∞ (D) = C∞ (M, A2m ) = C∞ (M) ⊗ A2m . Let βj k = (Xj∗ , Xk∗ ) be the structure σ : T ∗ M → T M is determined by m functions. Then the bundle map ∗ ∞ σ (Xk ) = j =1 βj k Xj . Thus for any f, g ∈ Cc (M) we have {f, g} =
βj k Xj (f )Xk (g),
1≤j,k≤m
1 1 π (f ) = f ⊗ 1 + √ 2 ( Xj (f ) ⊗ ϕ (ej ) + βj k Xk (f ) ⊗ ϕ (em+j )), 2 1≤j ≤m 1≤j,k≤m where ej and ϕ are as in Definition 2.1.
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Example 3.7. Let M be the m-torus Tm , and let x1 , · · · , xm be the standard coordinates. Let θ be a real skew-symmetric m × m matrix. Define a Poisson bracket {·, ·} on M by {f, g} = Let Xj =
∂ ∂xj
1 2
1≤j,k≤m
θj k
∂f ∂g . ∂xj ∂xk
. Let βj k and J be as in Example 3.6. Then βj k = 21 θj k . Thus
∂f 1 1 ∂f 1 π (f ) = f ⊗ 1 + √ 2 ( ⊗ ϕ (ej ) + θj k ⊗ ϕ (em+j )). ∂x 2 ∂x 2 j k 1≤j ≤m 1≤j,k≤m This is very different from Rieffel’s Moyal product approach [23, 26], which leads to the noncommutative torus Aθ . Example 3.8. Let g be a Lie algebra, and let M be the dual g∗ equipped with the LiePoisson bracket. Let v1 , · · · , vm be a basis of g, and let µ1 , · · · , µm be the dual basis of g∗ . Let cj kl be the structure constants satisfying [vj , vk ] = cj kl vl . We may take Xj in Example 3.6 to be µj . Then Xj∗ = vj , and βj k = l cj kl vl . Thus 1 1 π (f ) = f ⊗ 1 + √ 2 ( µj (f ) ⊗ ϕ (ej ) 2 1≤j ≤m + cj kl vl µk (f ) ⊗ ϕ (em+j )). 1≤j,k,l≤m
4. Homotopy Our construction in Theorem 3.3 depends on the choice of J . We define homotopy of strict quantizations first, then show that the homotopy class of our construction is independent of the choice of J (Proposition 4.3). The definition of homotopy of strict quantizations is similar to the usual definition of homotopy of homomorphisms between C ∗ -algebras: j
j
Definition 4.1. Let {A , π } be strict quantizations of (M, ) over I for A, where j = 0, 1. By a homotopy of these two strict quantizations, we mean a continuous field of C ∗ -algebras {A,t } over I × [0, 1] and linear maps π,t : A → A,t such that (1) the restriction of this field on I × {t} gives a strict quantization of (M, ) over A for each t ∈ [0, 1] in a uniform way, i.e. for all f, g ∈ A we have lim sup [π,t (f ), π,t (g)]/(i) − π,t ({f, g}) = 0,
→0 0≤t≤1
(2) for t = 0, 1 the restriction of this field gives the strict quantizations {A0 , π0 } and {A1 , π1 } respectively. Remark 4.2. One may also define a weaker notion of homotopy without requiring the convergence in (1) to be uniform. We adopt the stronger one because the homotopies we construct here all satisfy the uniform condition. Clearly homotopy is an equivalence relation between strict quantizations of M.
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Proposition 4.3. The homotopy class of the strict quantization in Theorem 3.3 does not depend on the choice of the compatible almost complex structure J on T ∗ M ⊕ T M. Proof. Let J0 and J1 be two compatible almost complex structures on T M ⊕ T M. Let < ·, · >0 and < ·, · >1 be the induced inner products. Let < ·, · >t = t < ·, · >1 +(1 − t) < ·, · >0 for 0 ≤ t ≤ 1. The canonical way of constructing compatible almost complex structure from a given inner product [5] is continuous. Thus we get a continuous family of compatible almost complex structures Jt on T ∗ M ⊕ T M. Let (Dt , ϕ,t ) be the corresponding quantization bundle and maps. Denote by c (T ∗ M ⊕ T M) the space of compactly supported sections of T M ⊕ T M. Then the sections (ϕ,t (f ))0≤t≤1 for 0 < ≤ 1 and f ∈ c (T ∗ M ⊕ T M) generate a continuous field of C ∗ -algebras over [0, 1] with fibre ∞ (Dt ) at t. Now it is easy to see that the strict quantizations in Theorem 3.3 associated with Jt ’s combine together to give a homotopy of the ones associated with J0 and J1 . For given (M, , A, I ) we don’t know whether there is only one homotopy class of strict quantizations. But this is the case when = 0 and I is an interval. Notice first that for = 0 there is a canonical trivial strict quantization over any I , namely A = C∞ (M) and π is the canonical inclusion of A into C∞ (M). Proposition 4.4. When = 0 and I is an interval, every strict quantization over I is homotopic to the canonical one. Proof. Let {A , ϕ } be a strict quantization. Define a map γ : I × [0, 1] → I by γ (, t) = t. Then the pull back of the field {A } under γ is a continuous field over I × [0, 1] with fibre A,t = At at (, t). For each f ∈ A let {π,t (f )} be the pull back of the section {π (f )} under γ , namely π,t (f ) = πt (f ). Then clearly this is a homotopy between the canonical strict quantization and {A , ϕ }. 5. Local Strict Quantizations There are two different meanings for a strict quantization to be local. The first one is an intuitive one, meaning that the construction is local in the sense that we construct strict quantizations for open subsets of M first, then gluing them together to get a strict quantization for M. This includes our construction in Theorem 2.10 and the construction in [19]. The second one means that the algebras and maps {A , π } are local in the sense that A ⊆ ∞ (D ) for some (upper-semi)continuous field of C ∗ -algebras D over M and the maps π : C∞ (M) = ∞ (M × C) → A → ∞ (D ) are fibrewise. Here we’ll concentrate on the second meaning. Let X be a locally compact Hausdorff space. Recall that an C∞ (X)-algebra is a C ∗ -algebra A with an injective nondegenerate homomorphism γ : C∞ (X) → M(A) such that γ (C∞ (X)) being contained in the center ZM(A) of the multiplier algebra M(A) [10]. This is equivalent to saying that A is the global section algebra of an uppersemicontinuous field of C ∗ -algebras over X [21]. Under this correspondence the fibre algebra of the field at x ∈ X is A/γ (Ix )A, where Ix = {h ∈ C∞ (X) : h(x) = 0}. This motivates our definition of local strict quantizations: Definition 5.1. Let {A , π } be a strict quantization of (M, ) on I . We call {A , π } local if each A is a C∞ (M)-algebra with γ : C∞ (M) → ZM(A ) such that π (f ) − γ (f ) → 0 as → 0 for every f ∈ A.
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Clearly the strict quantizations in Theorem 2.10 are local. But the ones in [19] are not. Proposition 5.2. Let {A , π } be a local strict quantization of (M, ) on I . Let f, g ∈ Asa . If π (f ), π (g) ∈ (A )sa for all ∈ I , then {f, g} = 0. Proof. Using the embeddings γ in Definition 5.1 we’ll identify C∞ (M) as a subalgebra of M(A ). Then π ({f, g})−{f, g} → 0 as → 0. Thus Definition 1.1(3) becomes lim [π (f ), π (g)]/(i) − {f, g} = 0.
→0
(4)
For x ∈ M let Ix = {h ∈ C∞ (M) : h(x) = 0}, and let A,x = A /Ix A . Let β,x : A → A,x be the quotient map. Notice that the identity of M(A,x ) is β,x (h) for any h ∈ C∞ (M) with h(x) = 1. Taking β,x on (4) we get lim [β,x (π (f )), β,x (π (g))]/(i) − {f, g}(x) = 0.
→0
(5)
Notice that [β,x (π (f )), β,x (π (g))]/(i) is a self-commutator, i.e. of the form [S ∗ , S] for some S (for instance S = (2)−1/2 β,x (π (f ))−i(2)−1/2 β,x (π (g))). It is known that self-commutators can’t be invertible [22, Corollary 1]. Thus {f, g}(x) = 0. Corollary 5.3. An almost Poisson manifold (M, ) admits a Hermitian local strict quantization if and only if = 0. Proof. Assume that = 0 and that M admits a Hermitian local strict quantization. Then we can find a covector field Y ∗ ∈ (T ∗ M) with σ (Y ∗ ) = 0, where σ is as in Lemma 3.2 for E = T ∗ M. We claim that there is a vector field X = 0 such that X(df ) = 0 for all f ∈ A. If σ (dg) = 0 for some g ∈ Asa , by Proposition 5.2 we may take X = σ (dg). Otherwise we may take X = σ (Y ∗ ). Let Z be a nonconstant integral curve of X. Then the restriction of every f ∈ A on Z is constant, which contradicts A being dense in C∞ (M). This proves the “only if” part. The “if” part is trivial. Remark 5.4. We don’t know when a local strict quantization is homotopic to a Hermitian strict quantization. As a comparison, a star product on a symplectic manifold is equivalent to a Hermitian one if and only if its characteristic class is Hermitian [20]. Also every Poisson manifold has Hermitian star products [4]. Thus the strict quantizations in Theorem 2.10 are not Hermitian unless = 0. In fact we can say more: Proposition 5.5. Let {A , π } be a local strict quantization of (M, ) on I . Identify C∞ (M) as a subalgebra of M(A ) via γ . Let τ (f ) = π (f ) − f for 0 < ≤ 1 and f ∈ A. Assume that A ∩ (τ (A) + (τ (A))∗ ) = {0}
(6)
for every 0 < ≤ 1. If this is a strict deformation quantization, then it is Hermitian and = 0. Proof. Let f ∈ A and 0 < ≤ 1. Then (π (f ))∗ = π (g) for some g ∈ A. Thus f ∗ + (τ (f ))∗ = g + τ (g). By our assumption f ∗ = g. Thus this strict quantization is Hermitian. By Corollary 5.3 = 0.
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Corollary 5.6. Let {A , π } be a local strict quantization of (M, ) on I with = 0. Let τ be as in Proposition 5.5 and assume that (6) holds for all 0 < ≤ 1. Then {A , π } can’t be restricted to a dense ∗-subalgebra of A to get a strict deformation quantization of (M, ). In particular, the strict quantizations in Theorem 2.10 can’t be restricted to a dense ∗-subalgebra of Cc∞ (M) to get a strict deformation quantization unless = 0. Proof. By Lemma 2.9 the strict quantizations in Theorem 2.10 satisfy (6).
Corollary 5.7. For any (M, ) there is a faithful flabby strict quantization, which can’t be restricted to any dense ∗-subalgebra of A to get a strict deformation quantization. Proof. The case = 0 follows from Theorem 3.3 and Corollary 5.6. The case = 0 is settled in [16]. Corollary 5.7 gives Question 25 in [27] a negative answer, which asks whether there is an example of a faithful strict quantization of a Poisson manifold such that it’s impossible to restrict it to some dense ∗-subalgebra to get a strict deformation quantization. This leaves the question whether we can require the strict quantization to be Hermitian.
6. Functorial Properties It is unlikely that there is a universal way to construct a canonical strict quantization for each Poisson manifold such that it gives a contravariant functor from the category of Poisson manifolds with (proper) Poisson maps to the category of continuous field of C ∗ -algebras over I [14]. Instead, Landsman proposed other categories closely related to Morita equivalence, and showed that there is such a functor on the subcategory of duals of integrable Lie algebroids. Though our construction in Theorem 3.3 doesn’t give a contravariant functor, it does have some properties similar to functors. In this section we discuss two questions: M (1) Fixing a strict quantization {AM , π } of (M, M ) on I , for any proper Poisson map φ from M to another almost Poisson manifold (N, N ) can we find a N strict quantization {AN , π } of (N,∗ N ) on I with a “homomorphism” of these two strict quantizations extending φ : C∞ (N ) → C∞ (M), i.e. a homomorphism M ξ : AN → A for each ∈ I such that these maps {φ } send continuous sections to continuous ones and ξ ◦ πN = πM ◦ φ ∗ ? (2) The similar question but fixing the strict quantization of N instead.
The first question has a positive answer because of Theorem 1.2 and the following proposition, whose proof is just routine verification. N N M Proposition 6.1. Let {AM , π } and {A , π } be strict quantizations of (M, M ) and M N (N, N ) on I for A and A respectively. Let φ : M → N be a proper Poisson map with N ∗ φ ∗ (AN ) ⊆ AM . Then the sections {(f, g) : f ∈ ({AM }), g ∈ ({A }), f0 = φ (g0 )} N M ∗ determine a continuous field of C -algebras over I with fibre A ⊕ A at = 0 and fibre C∞ (N ) at = 0. And (πM ◦ φ ∗ ) ⊕ πN for = 0 give a strict quantization for (N, N ) on I for AN .
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N N M Remark 6.2. (1) When both {AM , π } and {A , π } are local in the sense of DefiniN N M ∗ tion 5.1, so is {AM ⊕ A , (π ◦ φ ) ⊕ π }; (2) When (M, M ) = (N, N ) and φ = idM , Proposition 6.1 shows that the set of isomorphism classes of strict quantizations of (M, ) over I for A has a natural abelian semigroup structure. Clearly the addition is compatible with homotopy defined in Definition 4.1. Thus the set of homotopy classes of strict quantizations of (M, ) over I for A is also an abelian semigroup.
For the second question we have a partial positive answer: Proposition 6.3. Let (N, N ) be an almost symplectic manifold, and let k ≥ dim N . N Then there is a strict quantization {AN , π } of (N, N ) as constructed in Theorem 3.3 such that for any proper Poisson map φ : (M, M ) → (N, N ) with k ≥ dim M M there is a strict quantization {AM , π } of (M, M ) as constructed in Theorem 3.3 and N M homomorphisms ξ : A → A sending continuous sections to continuous ones with ξ ◦ πN = πM ◦ φ ∗ . Proof. Let n = dim N , and let n ≤ m ≤ k. We’ll choose a special asymptotic representation of Heisenberg commutation relations of dimension 2m. Let e1 , . . . , e2m and be the standard basis of R2m and R2k respectively. Then the linear map e1 , . . . , e2k 2m η : R → R2k defined by η(ej ) = ej , η(ej +m ) = ej +k for 1 ≤ j ≤ m preserves the standard symplectic structure and the standard compatible almost complex structure. Thus U (m) can be thought of as the subgroup of U (k) fixing el , el+k for m < l ≤ k. Let (A2k , ϕ ) be an asymptotic representation of Heisenberg commutation relations of dimension 2k. Let A2m be the C ∗ -subalgebra generated by ∪0<≤1 (ϕ ◦ η)(R2m ). Then (A2m , ϕ ◦ η) is an asymptotic representation of Heisenberg commutation relations of dimension 2m. Fix a compatible almost complex structure J N on T ∗ N ⊕ T N . Let x ∈ M. Since (N, N ) is almost symplectic, (T ∗ φ)φ(x) is injective. Thus n ≤ dim M. Let (D N , σ N , ϕN ) and σ M be as in Theorem 3.3 and Lemma 3.2 for N and M respectively. Since (N, N ) is almost symplectic, σ N is invertible. Then we have a linear map θx := N )−1 : T N ∗ ∗ σxM ◦ (T ∗ φ)φ(x) ◦ (σφ(x) φ(x) → T Mx . Let ζx := (T φ)φ(x) ⊕ θx : T Nφ(x) ⊕ ∗ T Nφ(x) → T Mx ⊕T Mx . Easy computation shows that T Mx ◦θx is the identity map on T Nφ(x) , and that ζx preserves the canonical symplectic structure on T ∗ Nx ⊕ T Nx . Then {ζx (T ∗ Nφ(x) ⊕T Nφ(x) )} is an almost symplectic subbundle of T ∗ M ⊕T M, which we’ll denote by E. Let Fx = (Ex )⊥ with respect to the almost symplectic structure. Then {Fx } is also an almost symplectic subbundle of T ∗ M ⊕T M, which we’ll denote by F . Clearly T ∗ M ⊕T M = E ⊕F , and J N induces a compatible almost complex structure J E on E. Take a compatible almost complex structure J F on F . Then J M := J E ⊕ J F is a compatible almost complex structure on T ∗ M ⊕ T M. Let (D M , ϕN ) be as in Theorem 3.3 for (M, M , J M ). Say m = dim M. Notice that ζx (u1 ), · · · , ζx (un ), µ1 , . . . , µm−n , ζx (v1 ), . . . , ζx (vn ), γ1 , . . . , γm−n is a unitary basis of T ∗ Mx ⊕ T Mx for any unitary basis u1 , · · · , un , v1 , · · · , vn of T ∗ Nφ(x) ⊕ T Nφ(x) and any unitary basis µ1 , · · · , µm−n , γ1 , · · · , γm−n of Fx . Because of our choice of A2m and A2n the map ζx determines a unital C ∗ -algebra embedding N ξx : Dφ(x) → DxM such that ϕM,x ◦ ζx = ξx ◦ ϕN,φ(x) for all 0 < ≤ 1. Since φ is
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proper, the ξx ’s combine to give a homomorphism ξ : ∞ (D N ) → ∞ (D M ) whose restriction on C∞ (N ) is ψ ∗ . Clearly ξ := ξ satisfy the requirement. Remark 6.4. (1) When both M and N have G-equivariant Riemannian metrics there is an obvious G-equivariant version of Proposition 6.3; (2) If we can find a C ∗ -algebra A∞ with an action of U (∞) := ∪n∈N U (n) and linear maps ϕn : R2n → A∞ for all n ∈ N compatible with the embedding η : R2n → R2k in the proof of Proposition 6.3 such that for each n these maps give an asymptotic representation of Heisenberg commutation relations, then we can use A∞ instead of A2k in the above proof and hence throw away the requirement k ≥ dim M. But we don’t know whether such infinite dimensional asymptotic representation of Heisenberg commutation relations exists or not.
7. Asymptotic Representation of Heisenberg Commutation Relations In this section we prove Theorem 2.3. Lemma 7.1. Let H be a separable Hilbert space with orthonormal basis {ej }∞ j =1 ∪ ∞ {ej }j =1 . Then there exist norm continuous paths T (), S() of operators in B(H ) for 0 < ≤ 1 such that (1) [T (), S()](e2j −1 ) = (1 + )ie2j −1 , (2) [T (), S()](e2j ) = (1 − )ie2j , (3) [T (), S()](ej ) = iej , (4) T () and S() are bounded uniformly in . Brown and Pearcy [3] proved that for a separable Hilbert space H an operator R ∈ B(H ) is a commutator if and only if it is not a non-zero scalar modulo compact operator. For the “if” part, their proof is constructive. Since we need T () and S() to be continuous, and want some control on their norms, and the construction in [3] depends on some choices of isomorphisms of Hilbert spaces, we write down the proof of Lemma 7.1 here, though it is just following the construction in [3]. Proof. Let ηj = √1 (e2j −1 + e2j ), ηj = √1 (e2j −1 − e2j ). Define R() ∈ B(H ) 2 2 by R()(e2j −1 ) = (1 + )ie2j −1 , R()(e2j ) = (1 − )ie2j and R()(ej ) = iej . Define Z() ∈ B(H ) by Z()(ηj ) = ηj , Z()(ei ) = ej and Z()(ηj ) = ηj + ηj . 11 Then on spanC {e2j −1 , e2j }, with ηj , ηj as basis, Z() and Z()−1 are and 0 −1 respectively. Therefore Z() < 3 and Z()−1 < 3−1 . Hence it −1 0 1 suffices to find continuous paths T (), S() such that [T (), S()] = Z()−1 R()Z() and T () , S() are bounded uniformly in . ∞ ∞ If we identify H with the closure span of {ei }∞ 1 , {ηi }1 , {ηi }1 respectively and hence identify H with H ⊕ H ⊕ H , simple calculation shows that I 0 0 2I I . Z()−1 R()Z() = i 0 0 (2 − 1)I 0
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If we identify H with the closure span of e1 , η1 , e2 , η2 , ...,and hence identify H with AW H ⊕ H , we see that Z()−1 R()Z() is of the form , where A, W don’t depend B 0 ∗ on and W is an isometry with dim(ker(W )) = ∞ and B is continuous for 0 < ≤ 1 with B ≤ 1. By Lemma 5.1 of [3] we can find X ∈ B(H ) such that A + W X = [B1 , B2 ] for some B1 , B2 ∈ B(H ) and XW = 0. Replacing B1 by some B1 + λI we may assume that B1 − I is invertible. By the similarity transformation I 0 AW I 0 A + WX W = , −X I B 0 XI −XA − XW X + B 0 it suffices to find continuous paths T (), S() such that A + WX W [T (), S()] = −XA − XW X + B 0 and T () , S() are bounded uniformly in . Simple calculation shows that B2 (B1 − I )−1 W B1 0 , S() = T () = 0 I (−XA − XW X + B)(I − B1 )−1 0 satisfy the requirements.
Lemma 7.2. Let H be a separable Hilbert space. Then for any n ∈ N there exist norm continuous paths T1 (), T2 (), · · · , Tn (), S1 (), S2 (), · · · , Sn () of the operator in B(H ) for 0 < ≤ 1 such that j
(1) lim→0 [Tj (), Sk ()] − δk i = 0, (2) [Tj (), Tk ()] = [Sj (), Sk ()] = 0, (3) −1/3 Tj () and −1/3 Sj () are bounded uniformly in . Proof. Let T () and S() be as in Lemma 7.1. Let Tj () = I ⊗ I ⊗ · · · ⊗ I ⊗ T (1/3 ) ⊗ I ⊗ · · · ⊗ I, Sj () = I ⊗ I ⊗ · · · ⊗ I ⊗ S(1/3 ) ⊗ I ⊗ · · · ⊗ I, where T (1/3 ), S(1/3 ) are at the j-th place. Identify H with H ⊗n . Then clearly Tj (), Sj () satisfy the conditions. Proof of Theorem 2.3. Let H and Tj (), Sj () be as in Lemma 7.2. For each 0 < ≤ 1 define a R-linear map φ : R2n → B(H ) by φ (ej ) = Tj (), φ (ej +n ) = Sj () for 1 ≤ j ≤ n, where e1 , · · · , e2n is the standard basis of R2n . Clearly {φ } satisfy the conditions (1)-(3) in Definition 2.1.
Denote the action of U (n) on R2n by σ . Consider the product C ∗ -algebra h∈U (n) B(H ), whose elements are bounded
maps f : U (n) → B(H ). There is a natural (discontinuous) action ρ of U (n) on h∈U (n) B(H ) given by ρg (f )(h) = f (g −1 h). For
each 0 < ≤ 1 define a R-linear map ϕ : R2n → h∈U (n) B(H ) by ϕ (u)(h) = 2n ∗ φ (σ
h−1 (u)) for u ∈ R . Clearly ϕ is U (n)-equivariant. Let A2n be the C -subalgebra of h∈U (n) B(H ) generated by ∪0<≤1 ϕ (R2n ). Then the restriction of ρ on A2n is continuous. Clearly ϕ = φ and ϕ − ϕ = φ − φ , which verify the
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conditions (2) and (3) of Definition 2.1. Using Lemma 7.2(2) simple calculation shows that max
1≤j
j
j
[ϕ (ej ), ϕ (ek )] − δk−n i ≤ max [Tj (), Sk ()] − δk i . 1≤j,k≤n
Then Definition 2.1(1) is also satisfied. In particular, when is small enough we have
[ϕ (e1 ), ϕ (en+1 )] − i < 1. Then [ϕ (e1 ), ϕ (en+1 )] is invertible in h∈U (n) B(H )
and hence A2n contains the identity of h∈U (n) B(H ). Acknowledgements. I would like to thank Marc Rieffel for many helpful discussions and suggestions. I also thank Henrique Bursztyn for valuable discussions about deformation quantizations.
References 1. Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization, I, II. Ann. Phys. 111.1 , 61–110, 111–151 (1978) 2. Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of K¨ahler manifolds and gl(N), N → ∞ limit. Commun. Math. Phys. 165.2 , 281–296 (1994) 3. Brown, A., Pearcy, C.: Structure of commutators of operators. Ann. of Math. 82, 112–127 (1965) 4. Bursztyn, H., Waldmann, S.: On positive deformations of ∗-algebras. In: Conf´erence Mosh´e Flato 1999, Vol. II (Dijon), Math. Phys. Stud. no. 22, Dordrecht: Kluwer Acad. Publ., 2000, pp. 69–80 5. Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics no. 1764. Berlin: Springer-Verlag, 2001 6. Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Berkeley Mathematics Lecture Notes no. 10. Providence, RI: American Mathematical Society, Berkeley, CA: Berkeley Center for Pure and Applied Mathematics, 1999 7. De Wilde, M., Lecomte, P. B.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary sympletic manifolds. Lett. Math. Phys. 7.6, 487–496 (1983) 8. Dixmier, J.: C*-algebras Translated from the French by Francis Jellett. North-Holland Mathematical Library, Vol. 15. Amsterdam-New York-Oxford: North-Holland Publishing Co., 1977 9. Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79 , 59–103 (1964) 10. Kasparov, G.G.: Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91.1, 147–201 (1988) 11. Klimek, S., Lesniewski, A.: Quantum Riemann surfaces for arbitrary Planck’s constant. J. Math. Phys. 37.5, 2157–2165 (1996) 12. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66.3, 157–216 (2003) 13. Landsman, N.P.: Mathematical Topics between Classical and Quantum Mechanics. Springer Monographs in Mathematics. New York: Springer-Verlag, 1998 14. Landsman, N.P.: Quantization as a functor. In: Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math. no. 315, Providence, RI: Am. Math. Soc., 2002, pp. 9–24 15. Landsman, N.P., Ramazan, B.: Quantization of Poisson algebras associated to Lie algebroids. In: Groupoids in analysis, geometry, and physics (Boulder, CO, 1999), Contemp. Math. no. 282, Providence, RI: Am. Math. Soc., 2001, pp. 159–192 16. Li, H.: Flabby strict deformation quantizations and K-groups. K-Theory to appear. 17. Natsume, T.: C ∗ -algebraic deformation quantization of closed Riemann surfaces. In: C ∗ -algebras (Munster, 1999), Berlin: Springer, 2000, pp. 142–150 18. Natsume, T., Nest, R.: Topological approach to quantum surfaces. Commun. Math. Phys. 202.1 , 65–87 (1999) 19. Natsume, T., Nest, R., Peter, I.: Strict quantizations of symplectic manifolds. Lett. Math. Phys. 66, 73–89 (2003) 20. Neumaier, N.: Local ν-Euler Derivations and Deligne’s Characteristic Class of Fedosov Star Products and Star Products of Special Type. Commun. Math. Phys. 230, 271–288 (2002) 21. Nilsen, M.: C ∗ -bundles and C0 (X)-algebras. Indiana Univ. Math. J. 45.2, 463–477 (1996) 22. Radjavi, H.: Structure of A∗ A − AA∗ . J. Math. Mech. 16, 19–26 (1966) 23. Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122.4, 531–562 (1989)
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24. Rieffel, M.A.: Deformation quantization and operator algebras. In: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), Proc. Sympos. Pure Math., 51, Part 1, Providence, RI: Am. Math. Soc., 1990, pp. 411–423 25. Rieffel, M.A.: Lie group convolution algebras as deformation quantizations of linear Poisson structures. Am. J. Math. 112, 657–686 (1990) 26. Rieffel, M.A.: Deformation Quantization for Actions of Rd . Mem. Amer. Math. Soc. no. 506. Providence, RI: Amer. Math. Soc., 1993 27. Rieffel, M.A.: Questions on quantization. In: Operator algebras and operator theory (Shanghai, 1997), Providence, RI: Am. Math. Soc., 1998, pp. 315–326 Communicated by A. Connes
Commun. Math. Phys. 257, 273–285 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1225-1
Communications in
Mathematical Physics
New Infinite Series of Einstein Metrics on Sphere Bundles from AdS Black Holes Yoshitake Hashimoto1 , Makoto Sakaguchi2 , Yukinori Yasui3 1
Department of Mathematics, Osaka City University, Sumiyoshi, Osaka 558-8585, Japan. E-mail: [email protected] Osaka City University Advanced Mathematical Institute (OCAMI), Sumiyoshi, Osaka, 558-8585, Japan. E-mail: [email protected] 3 Department of Physics, Osaka City University, Sumiyoshi, Osaka, 558-8585, Japan. E-mail: [email protected]
2
Received: 1 April 2004 / Accepted: 11 May 2004 Published online: 25 November 2004 – © Springer-Verlag 2004
Abstract: A new infinite series of Einstein metrics is constructed explicitly on S 2 × S 3 , and the non-trivial S 3 -bundle over S 2 , containing infinite numbers of inhomogeneous ones. They appear as a certain limit of 5-dimensional AdS Kerr black holes. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S d−2 -bundle over S 2 from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on CP 2 CP 2 . 1. Introduction Anti-de Sitter (AdS) spaces have attracted renewed interest after the AdS/CFT correspondence conjecture [1], which relates the properties of the supergravity on AdS and those of the strongly coupled gauge theory on the AdS boundary. For example, the Hawking-Page phase transition [2] between AdS and AdS Schwarzschild black holes was interpreted [3] as the phase transition between confining and deconfining phases of the dual gauge theory. Motivated by this, the study of AdS black holes has been extended in various directions. Among them, AdS Kerr black holes with two angular momenta in 5-dimensions, as well as ones with one angular momentum in d-dimensions were constructed in [4]. On the other hand, an inhomogeneous Einstein metric on CP 2 CP 2 was constructed by Page [5]. This metric is of cohomogeneity one with principal orbits S 3 . It was obtained as a certain limit of the 4-dimensional de Sitter black hole together with the Wick rotation. It should be emphasized that this metric is the first example of inhomogeneous Einstein metrics. Furthermore, B¨ohm proved the existence of an infinite series of Einstein metrics of cohomogeneity one with positive scalar curvature on S N (5 ≤ N ≤ 9) and S N1 +1 × S N2 (5 ≤ N1 + N2 + 1 ≤ 9, N1 > 1, N2 > 1) [6].
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Combining these two observations, we explicitly construct new Einstein metrics with positive scalar curvature on sphere bundles, applying the method developed in [5] to the 5-dimensional AdS Kerr black hole with two angular momenta and the d-dimensional AdS Kerr black hole with one angular momentum constructed in [4]. In summary, we will construct S 3 (the non-trivial S 3 -bundle • an infinite series of Einstein metrics on S 2 ×S 3 and S 2 × 2 over S ) parameterized by a pair of integers (k1 , k2 ). The bundle type depends on the parity of k1 + k2 ; it is trivial if k1 + k2 is even, and is non-trivial otherwise. When k1 = k2 , the metrics are inhomogeneous (see Theorem 1). When k1 = k2 , they are homogeneous Einstein metrics on S 2 × S 3 (see Theorem 2). S d−2 , the non-trivial S d−2 -bundle over • an inhomogeneous Einstein metric on S 2 × 2 S (see Theorem 3). S 3 in Theorem 1 obviously are not It should be noticed that the metrics on S 2 × included in the case of B¨ohm’s existence theorem and that the metrics on S 2 × S 3 are apparently different from the ones which are proved to exist in [6]. The metrics in The1,1 orem 2 coincide with the homogeneous Einstein metrics on Mk,1 , circle bundles over 1 1 CP × CP studied by Wang and Ziller [7]. The metric in Theorem 3 is of cohomogeneity one with principal orbits S 3 × S d−4 if d ≥ 5. In the case of d = 4, it reproduces S 2 = CP 2 CP 2 with principal orbits S 3 . the Page metric on S 2 × This paper is organized as follows. In Sect. 2, we apply the method [5] to the 5dimensional AdS Kerr black hole with two angular momenta, and obtain the first class of the Einstein metrics. We will derive the second class of the Einstein metrics from the d-dimensional AdS Kerr black hole [4] with one angular momentum in Sect. 3. 2. 5-dimensional Einstein Metrics The 5-dimensional Euclidean de Sitter (dS) Kerr metric may be extracted from the Lorentzian AdS Kerr metric [4] by the substitution t → −iτ, a → −iα, b → −iβ with l → −il (AdS → dS), 2 2 α sin2 θ r 2 − α2 r β cos2 θ θ sin2 θ αdτ + dφ − dψ + dφ g5 = 2 dτ − ρ α β ρ2 α 2 r2 − β2 θ cos2 θ βdτ + + dψ β ρ2 2 β(r 2 − α 2 ) sin2 θ 1 − r 2l2 α(r 2 − β 2 ) cos2 θ − 2 2 αβdτ + dφ + dψ r ρ α β +
ρ2 2 ρ2 2 dr + dθ , r θ
(2.1)
where ρ 2 = r 2 − α 2 cos2 θ − β 2 sin2 θ, 1 r = 2 (r 2 − α 2 )(r 2 − β 2 )(1 − r 2 l 2 ) − 2M, r θ = 1 − α 2 l 2 cos2 θ − β 2 l 2 sin2 θ
(2.2)
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275
with parameters α = 1 − α 2 l 2 and β = 1 − β 2 l 2 . The radii of the horizons are given by the roots of r = 0. First, we consider an extremal case. If there is a double root r0 , the number of the free parameters reduces. The parameters r0 , M, α and β are written in terms of l and the dimensionless parameters ν1 = α/r0 ,
ν2 = β/r0 ,
(2.3)
as1 r0 = M0 =
1 − ν12 ν22
1/2
2 − ν12 − ν22
l −1 ,
(1 − ν12 )2 (1 − ν22 )2 (1 − ν12 ν22 ) 2(2 − ν12 − ν22 )2
0α = 1 − 0β = 1 −
ν12 (1 − ν12 ν22 ) 2 − ν12 − ν22 ν22 (1 − ν12 ν22 ) 2 − ν12 − ν22
l −2 ,
, (2.4)
.
In terms of these parameters, we have ˜ r = −(r − r0 )2 (r), 1 ˜ (r) = 2 (r + r0 )2 (l 2 r 2 − ν12 ν22 ). r
(2.5)
Let us consider a nearly extremal case; it has two horizons located at r = r1 ≡ r0 − ε and r = r2 ≡ r0 + ε, where the parameter ε represents a small deviation from the case of the double root. In the region between two horizons, (ε) = {r|r1 ≤ r ≤ r2 }, we introduce a new radial coordinate χ (0 ≤ χ ≤ π) by r = r0 − ε cos χ .
(2.6)
˜ restricted to (ε) is then given by The r = −(r − r1 )(r − r2 )(r) ˜ 0 ) sin2 χ + O(ε 3 ), r = ε2 (r
(2.7)
where ˜ 0) = (r
4(1 + ν12 ν24 + ν14 ν22 − 3ν12 ν22 ) 2 − ν12 − ν22
.
(2.8)
We consider the limit ε → 0 of the metric (2.1) in the region (ε). In this limit, the term proportional to r vanishes and so leads to a singularity of the metric. To avoid this, we define a rescaled time coordinate η= 1
˜ 0) ε (r τ 2 r0 (1 − ν12 )(1 − ν22 )
The suffix 0 is added for the case of the double root.
+ O(ε 2 ).
(2.9)
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It is also convenient to define the azimuthal angles ββ αα τ, φ2 = ψ + 2 τ. r12 − α 2 r1 − β 2
φ1 = φ +
(2.10)
The new coordinates (η, χ , φ1 , φ2 , θ) are then well-behaved local coordinates in the limit ε → 0: 2 ρ02 r α sin2 θ β cos2 θ (a) 2 dτ − sin2 χ dη2 , dφ − dψ → (2.11) ˜ 0) ρ α β (r 2 r 2 − α2 θ sin2 θ αdτ + (b) dφ ρ2 α 2 4 2 2 r0 (1 − ν1 ) 0 2 dφ1 4ν1 (1 − ν22 ) 2 χ sin → θ sin θ − (2.12) dη , ˜ 0) 0α 2 ρ02 (1 − ν12 )(r 2 r2 − β2 θ cos2 θ βdτ + dψ (c) ρ2 β 2 r 4 (1 − ν22 )2 0 4ν2 (1 − ν12 ) dφ2 2 2 χ → 0 cos θ − , (2.13) dη sin θ ˜ 0) 2 ρ02 0β (1 − ν22 )(r 2 1 − r 2l2 β(r 2 − α 2 ) sin2 θ α(r 2 − β 2 ) cos2 θ (d) αβdτ + dφ + dψ r 2ρ2 α β 4 2 2 2 r (1 − ν1 )(1 − ν2 ) ν2 (1 − ν1 ) 2 ν1 (1 − ν22 ) → 02 sin θ dφ + cos2 θdφ2 1 0α ρ0 (2 − ν12 − ν22 ) 0β 2 4ν1 ν2 ρ02 2 χ dη − sin , (2.14) ˜ 0 )r 2 2 (r 0 ρ02 ρ2 2 dr → dχ 2 , ˜ 0) r (r ρ2 ρ2 2 (f ) dθ → 00 dθ 2 , θ θ
(2.15)
(e)
(2.16)
where ρ02 = r02 (1 − ν12 cos2 θ − ν22 sin2 θ ), 0θ = 1 −
1 − ν12 ν22 (ν12 cos2 θ 2 − ν12 − ν22
+ ν22 sin2 θ).
(2.17) (2.18)
Finally, we obtain a metric with two parameters ν1 and ν2 g = h2 (θ )dθ 2 +
2
aij (θ )ωi ⊗ ωj + b2 (θ )gS 2 .
(2.19)
i,j =1
Here, gS 2 represents the standard metric on S 2 , gS 2 = dχ 2 + sin2 χ dη2 .
(2.20)
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277
The 1-forms ωi (i = 1, 2) are defined by ωi = dψi + ki cos χ dη,
(2.21)
where k1 = k2 =
ν1 (1 − ν22 )(2 − ν22 − ν12 ν22 ) 4ν1 (1 − ν22 )0α = , ˜ 0) 1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22 (1 − ν12 )(r 4ν2 (1 − ν12 )0β ˜ 0) (1 − ν22 )(r
=
ν2 (1 − ν12 )(2 − ν12 − ν12 ν22 ) 1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22
,
(2.22)
and ψi are introduced as φi =
1 (ψi + ki η). 2
(2.23)
The metric components are found to be2 h2 =
1 − ν12 cos2 θ − ν22 sin2 θ
a22
a12
,
1 − ν12 2 sin2 θ = 20α 1 − ν12 cos2 θ − ν22 sin2 θ ν22 (1 − ν12 )(1 − ν22 ) 2 0 sin θ , × θ − 2 − ν12 − ν22 1 − ν22 2 cos2 θ = 0 2 2β 1 − ν1 cos2 θ − ν22 sin2 θ ν12 (1 − ν12 )(1 − ν22 ) 0 2 × θ − cos θ , 2 − ν12 − ν22 ν1 ν2 (1 − ν12 )2 (1 − ν22 )2 sin2 θ cos2 θ , =− 0 2 2 2 0 4α β (2 − ν1 − ν2 ) 1 − ν1 cos2 θ − ν22 sin2 θ
a11
0θ
b2 =
1 − ν12 cos2 θ − ν22 sin2 θ . ˜ 0) (r
(2.24)
(2.25)
(2.26) (2.27) (2.28)
It is straightforward to calculate the Ricci curvature. We find that (2.19) is the Einstein metric with the scalar curvature 20(1 − ν12 ν22 )/(2 − ν12 − ν22 ). In the following, we shall consider three cases for two parameters ν1 and ν2 : Case A. ν12 , ν22 > 1 and ν12 = ν22 , Case B. 0 ≤ ν12 , ν22 ≤ 1 and ν12 = ν22 , Case C. ν12 = ν22 ≡ ν. 2 The metric is symmetric with respect to the exchange (ψ , ν ) ↔ (ψ , ν ). Taking account of this 1 1 2 2 symmetry, we often discuss only the one side.
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These conditions ensure that singularities of the Riemannian curvature disappear, and further the metric components are non-negative, i.e., the eigenvalues of aij are nonnegative and h2 , b2 > 0. Next, we consider the condition to avoid orbifold singularities, which restricts the range of the angles (θ, ψi ) and the parameters ki . We calculate the determinant 2 sin2 θ cos2 θ0θ (1 − ν12 )(1 − ν22 ) . (2.29) det(aij ) = 40α 0β 1 − ν12 cos2 θ − ν22 sin2 θ Now clearly θ = 0, π/2 are zeros of the determinant and so the range of θ must be restricted to 0 ≤ θ ≤ π/2. The singularities at θ = 0, π/2 are removable bolt singularities. Indeed, near the boundaries the metric behaves as g→ and g→
1 − ν22 0β
1 − ν12 1 (dθ 2 + θ 2 dψ12 ) + gLk2 for θ → 0, 0 α 4 π 1 π π 2 2 2 d( − θ ) + ( − θ ) dψ2 + gLk1 for θ → , 2 4 2 2
(2.30)
(2.31)
where gLk1 = a11 (π/2)(dψ1 + k1 cos χ dη)2 + b2 (π/2)gS 2 ,
(2.32)
gLk2 = a22 (0)(dψ2 + k2 cos χ dη) + b (0)gS 2 .
(2.33)
2
2
Thus these singularities are removable provided that the ranges of ψi (i = 1, 2) are chosen to be 0 ≤ ψi ≤ 4π . In this range, (θ, ψ1 /2) as θ → 0 and (π/2 − θ, ψ2 /2) as θ → π/2 are the usual polar coordinates on R2 . We also demand that ki is integral (see Eq.(2.22)), then the 1-forms ωi = dψi + ki cos χ dη are identified with connections on the lens spaces Lki = L(ki , 1) = S 3 /Zki , and each ki represents the first Chern number (or the monopole charge) as a circle bundle on S 2 . This yields that the manifolds near the boundaries are R2 × Lki , which collapses onto {point} × Lk at the boundaries. Remark 1. There exists a nontrivial root of 0θ = 0, θ = θ0 ,
cos2 θ0 =
2 − ν12 − 2ν22 + ν12 ν24 (1 − ν12 ν22 )(ν12 − ν22 )
(2.34)
other than θ = 0 and π/2. In order to avoid curvature singularities, we restrict θ0 to be in the region 0 < θ0 < π/2 and the parameters (ν1 , ν2 ) in the following region: Case D. ν22 > 1, ν22 (1 + ν12 ) < 2 or ν22 < 1, ν22 (1 + ν12 ) > 2, Case E. ν12 > 1, ν12 (1 + ν22 ) < 2 or ν12 < 1, ν12 (1 + ν22 ) > 2. Then, the curvature is finite if we choose the range as 0 ≤ θ ≤ θ0 for the Case D, and θ0 ≤ θ ≤ π/2 for the Case E. However one can not resolve the orbifold singularity at the boundary θ = θ0 like the Cases A, B and C. Having constructed the Einstein metric locally, we now proceed to the global issue for the Cases A, B and C. It is known that there are two inequivalent classes of S N -bundles
New Infinite Series of Einstein Metrics
279
over S 2 . The S N -bundles over S 2 = D+ ∪ D− (D± denote hemispheres) are obtained by attaching D+ × S N and D− × S N as (x, ξ ) ∼ (x, γ (x)ξ ), x ∈ D+ ∩ D− = S 1 , γ : S 1 → SO(N + 1).
(2.35)
They are classified by [γ ] ∈ π1 (SO(N + 1)) = Z2 . Globally our metrics (2.19) can be regarded as those on S 3 -bundles over S 2 . We will discuss the Cases A, B and C separately below. Case A. We can write the metric (2.19) with (2.20)–(2.28) as 2
gν1 ν2 = h2 (θ )dθ 2 +
aij (θ )ωi ⊗ ωj + b2 (θ )gS 2 ,
(2.36)
i,j =1
where h2 = a11
a22
1 − ν12 cos2 θ − ν22 sin2 θ
, (2.37) 1 − µ21 cos2 θ − µ22 sin2 θ 2 (1 − µ21 cos2 θ − ν22 sin2 θ) sin2 θ 1 ν1 (1 − ν22 )(2 − ν12 − ν22 ) ,(2.38) = 4 1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22 1 − ν12 cos2 θ − ν22 sin2 θ 2 (1 − ν12 cos2 θ − µ22 sin2 θ) cos2 θ 1 ν2 (1 − ν12 )(2 − ν12 − ν22 ) ,(2.39) = 4 1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22 1 − ν12 cos2 θ − ν22 sin2 θ
a12 = − b2 =
ν12 ν22 (1 − ν12 )2 (1 − ν22 )2 (2 − ν12 − ν22 )
sin2 θ cos2 θ , (2.40) − ν22 sin2 θ
4(1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22 )2 1 − ν12 cos2 θ (2 − ν12 − ν22 )(1 − ν12 cos2 θ − ν22 sin2 θ ) 4(1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22 )
(2.41)
with µ21 =
ν12 (1 − ν12 ν22 ) 2 − ν12
− ν22
,
µ22 =
ν22 (1 − ν12 ν22 ) 2 − ν12 − ν22
.
(2.42)
The metric (2.36) can be regarded as one on the associated S 3 -bundle of the principal T 2 -bundle over S 2 with the Euler classes (k1 , k2 ) ∈ H 2 (S 2 , Z)⊕2 = Z ⊕ Z. The invariant [γ ] ∈ Z2 of the S 3 -bundle is given by [γ ] = k1 + k2 mod 2.
(2.43)
The connection ω = ω1 ⊕ ω2 is given by ωi = dψi + cos χ dη,
0 ≤ ψi ≤ 4π/|ki |,
where we have rescaled the torus angles as ψi → ki ψi .
(2.44)
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Summarizing the consideration above, we state Theorem 1. Let ν1 and ν2 be real numbers in the region ν12 , ν22 > 1 and ν12 = ν22 together with the integral conditions; k1 = k2 =
ν1 (1 − ν22 )(2 − ν22 − ν12 ν22 ) 1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22 ν2 (1 − ν12 )(2 − ν12 − ν12 ν22 ) 1 + ν14 ν22 + ν12 ν24 − 3ν12 ν22
,
(2.45)
,
(2.46)
where (k1 , k2 ) ∈ Z ⊕ Z. Then, {gν1 ν2 } gives an infinite series of inhomogeneous Einstein metrics with positive scalar curvature 20(1 − ν12 ν22 )/(2 − ν12 − ν22 ) on S 3 -bundles over S 2 . If the integer k1 + k2 is even (odd), then the bundle is trivial (non-trivial). Remark 2. The region ν1 > 1, ν2 > 1 is mapped diffeomorphically onto the region k1 > 0, k2 > 0, k1 + k2 > 2 by (2.45) and (2.46). Hence, there exists the unique pair (ν1 , ν2 ) for each (k1 , k2 ) ∈ S, S = {(k1 , k2 ) ∈ Z ⊕ Z | k1 = ±k2 , k1 = 0, k2 = 0}. For example, they are numerically evaluated as (i) (k1 , k2 ) = (1, 2), (ν1 , ν2 ) = (3.31133, 2.14921), (ii) (k1 , k2 ) = (1, 3), (ν1 , ν2 ) = (7.68872, 3.06769), (iii) (k1 , k2 ) = (2, 3), (ν1 , ν2 ) = (5.85109, 4.13646). Remark 3. In the limit (ν1 , ν2 ) = (ν1 , ∞), the metric tends to sin2 θ cos2 θ sin2 θ (dψ1 + cos χ dη)2 + dψ22 + gS 2 , (2.47) 4 4 4 after the rescaling ν12 g → g. When we make a modification of the range of ψ1 , gν1 ∞ = dθ 2 +
0 ≤ ψ1 ≤ 4π/|k1 | −→ 0 ≤ ψ1 ≤ 4π,
(k1 =
1 + ν12 ), ν1
(2.48)
then (2.47) represents the standard metric on S 5 . Case B. The parameters ν1 and ν2 are restricted to 0 ≤ ν12 , ν22 ≤ 1 and ν12 = ν22 . By (2.22) we find that 0 ≤ |ki | ≤ 2 (i = 1, 2), hence there are two possibilities for (|k1 |, |k2 |) under the integral condition3 : (i) (|k1 |, |k2 |) = (1, 0), and (ii) (|k1 |, |k2 |) = (2, 0). The case (ii) corresponds to (ν1 , ν2 ) = (±1, ν2 ). The corresponding Einstein metric (2.19) is independent of ν2 and coincides with the standard S 5 -metric after a modification of the angle ψ1 = 2ψ˜ 1 (0 ≤ ψ˜ 1 ≤ 4π): 1 2 1 1 sin θ (d ψ˜ 1 + cos χ dη)2 + cos2 θdψ22 + sin2 θgS 2 . (2.49) 4 4 4 One can show that the analysis remains true even if the range is extended to ν2 > 1. In the case (i), we have (ν1 , ν2 ) = (± 21 , 0). Then, the Einstein metric (2.19) is of the form 2 1 − 17 cos2 θ 1 − 41 cos2 θ 2 7 + g= dθ sin2 θ(dψ1 + cos χ dη)2 16 1 − 41 cos2 θ 1 − 17 cos2 θ 1 1 7 + cos2 θ dψ22 + (1 − cos2 θ )gS 2 , (2.50) 4 16 4 g = dθ 2 +
3
Remember that the metric is symmetric under (ψ1 , ν1 ) ↔ (ψ2 , ν2 ).
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281
which gives a metric of cohomogeneity one with principal orbits S 3 × S 1 . The principal orbits collapse to S 2 × S 1 at θ = 0 and S 3 × {point} at θ = π/2. Globally the metric can be regarded as one on the non-trivial S 3 -bundle over S 2 . In section 2, we will construct Einstein metrics on S N -bundles over S 2 (see Theorem 3), generalizing the metric (2.50) to higher dimensions. Case C. When we put ν1 = ν2 ≡ ν, the metric (2.19) is4 2
gν = dθ 2 +
aij (θ )ωi ⊗ ωj +
i,j =1,2
2 + ν2 g 2, 4(2ν 2 + 1) S
(2.51)
where sin2 θ (2 + ν 2 cos2 θ), 4(2 + ν 2 ) cos2 θ = (2 + ν 2 sin2 θ), 4(2 + ν 2 ) ν2 sin2 θ cos2 θ. =− 4(2 + ν 2 )
a11 =
(2.52)
a22
(2.53)
a12
(2.54)
The 1-form ω1 ⊕ ω2 is a connection on the T 2 -bundle over S 2 , locally written as ωi = dψi + k cos χ dη
(0 ≤ ψi ≤ 4π ).
(2.55)
Here the coefficient k is evaluated as k=
ν(ν 2 + 2) , 2ν 2 + 1
(2.56)
by using (2.22), and it is required to be k ∈ Z. In this case the T 2 -bundles collapse to the same lens space Lk at each boundary. Notice that by introducing the Maurer-Cartan forms of SU(2), ψ 1 + ψ2 ψ1 + ψ2 σ1 = 2 cos dθ − sin sin θ cos θ(dψ1 − dψ2 ), (2.57) 2 2 ψ1 + ψ 2 ψ 1 + ψ2 σ2 = −2 sin dθ − cos sin θ cos θ(dψ1 − dψ2 ), (2.58) 2 2 σ3 = sin2 θ dψ1 + cos2 θ dψ2 ,
(2.59)
the fiber metric of (2.51) can be rewritten as gF = dθ + 2
2
aij (θ )dψi ⊗ dψj =
i,j =1
which reveals the SU(2) isometry of the metric. 4
2 We have rescaled the metric as 2+ν 2 g → g.
1 2 1 (σ + σ22 ) + σ 2, 4 1 2(2 + ν 2 ) 3
(2.60)
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Theorem 2. Let νk be real numbers satisfying k = νk (νk2 +2)/(2νk2 +1) ∈ Z. Then, {gνk } gives an infinite series of homogeneous Einstein metrics with positive scalar curvature 20(1 + νk2 )/(2 + νk2 ) on S 2 × S 3 . Proof. By the coordinate transformation, α = 2θ,
β=
1 1 (ψ2 − ψ1 ) and t = (ψ1 + ψ2 ), 2 2
(2.61)
the metric takes the form gνk =
1 2 + ν2 (dχ 2 + sin2 χ dη2 ) (dα 2 + sin2 αdβ 2 ) + 4 4(2ν 2 + 1) 1 + (dt + cos αdβ + k cos χ dη)2 , 2(2 + ν 2 )
(2.62)
1,1 of the circle bundle which represents a Kaluza-Klein metric on the total space Mk,1 over CP 1 × CP 1 with Euler class e = kα1 + α2 , where α1 and α2 are generators in 1,1 H 2 (CP 1 × CP 1 ; Z) = Z ⊕ Z. The space Mk,1 is diffeomorphic to S 2 × S 3 [7].
Remark 4. There exists the unique real number νk for each k ∈ Z. The value of νk is explicitly given by 1 1/3 [a − 8(3 − 2k 2 )a −1/3 + 4k], 6 a = −36k + 64k 3 + 12 96 − 183k 2 + 96k 4 ,
νk =
for k ≥ 0, and ν−k = −νk . Remark 5. In the case ν0 = 0, the metric coincides with the product metric on S 2 × S 3 : g0 = dθ 2 +
1 2 1 1 sin θ dψ12 + cos2 θdψ22 + gS 2 . 4 4 2
(2.63)
On the other hand, in the limit νk → ±∞ (k → ±∞), the fiber S 1 of S 2 × S 3 → × S 2 collapses, and the metric tends to the product Riemannian metric on S 2 × S 2 which is not Einstein: sin θ cos θ 2 1 2 g∞ = dθ 2 + dψ− + gS 2 (2.64) 2 8 S2
with ψ− = ψ1 − ψ2 . 3. d-Dimensional Einstein Metrics The AdS Kerr black hole in d-dimensions (d ≥ 4) was constructed in [4]. It can be straightforwardly transformed to the Euclidean form, gd =
r α θ sin2 θ r 2 − α2 dφ)2 (dτ − sin2 θ dφ)2 + (αdτ + 2 2 ρ ρ ρ2 ρ2 2 + dr 2 + dθ + r 2 cos2 θgS d−4 , r θ
(3.1)
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283
where gS d−4 is the standard metric on S d−4 with the positive scalar curvature (d −4)(d − 5), which is Einstein5 , and ρ 2 = r 2 − α 2 cos2 θ, r = (r 2 − α 2 )(1 − l 2 r 2 ) − 2Mr 5−d , θ = 1 − α 2 l 2 cos2 θ
(3.2) (3.3) (3.4)
with the parameter = 1 − α 2 l 2 . We find that there exists a double root r0 of r = 0, when the following condition for the parameters is satisfied: 1/2 d − 3 − (d − 5)ν 2 r0 = l −1 , d − 1 − (d − 3)ν 2 d−3 d − 3 − (d − 5)ν 2 2 −(d−3) (1 − ν)2 M0 = l , d − 1 − (d − 3)ν 2 d − 1 − (d − 3)ν 2
0 = 1 −
ν 2 (d − 3 − (d − 5)ν 2 ) , d − 1 − (d − 3)ν 2
(3.5) (3.6) (3.7)
where we have introduced a dimensionless parameter ν = α/r0 . Then, r takes the form ˜ r = −(r − r0 )2 (r), 1 ˜ (r) = d−5 (c0 + c1 r + · · · + cd−3 r d−3 ), r
(3.8) (3.9)
where 2(1 + i)(1 − ν 2 )2 d−i−5 r (0 ≤ i ≤ d − 6), d − 1 − (d − 3)ν 2 0 2d − 8 − 3(d − 5)ν 2 + (d − 5)ν 4 = , d − 1 − (d − 3)ν 2 = 2r0 l 2 ,
ci = cd−5 cd−4
cd−3 = l . 2
(3.10) (3.11) (3.12) (3.13)
The remaining procedure is completely parallel to the one in Sect. 2; consider a nearly extremal case, and take the limit ε → 0. Actually we define new coordinates (η, χ , φ1 ) instead of (τ, r, φ) as r = r0 − ε cos χ , ˜ 0) ε (r η= 2 τ + O(ε 2 ), r0 (1 − ν 2 ) α φ1 = φ + 2 (r1 ≡ r0 − ε), r1 − α 2
(3.14) (3.15) (3.16)
5 The sphere metric g S d−4 can be replaced by gMd−4 , where gMd−4 is an arbitrary Einstein metric on a (d − 4)-dimensional manifold with the positive scalar curvature (d − 4)(d − 5).
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˜ 0 ) can be calculated by (3.9); where (r ˜ 0) = (r
(d − 5)(d − 3)ν 4 − 2(d − 1)(d − 5)ν 2 + (d − 1)(d − 3) . d − 1 − (d − 3)ν 2
(3.17)
In the limit ε → 0, we find a one-parameter family of d-dimensional Einstein metrics gν = h (θ )dθ + 2
2
3
ai (θ )σ i ⊗ σ i , +b2 (θ )gS d−4 ,
(3.18)
i=1
with the positive scalar curvature d(d − 1)(d − 3 − (d − 5)ν 2 )/(d − 1 − (d − 3)ν 2 ). Here σ i (i = 1, 2, 3) are 1-forms defined by σ 1 = cos ψdχ + sin ψ sin χ dη, σ 2 = − sin ψdχ + cos ψ sin χ dη, σ 3 = dψ + k cos χ dη,
(3.19) (3.20) (3.21)
with φ1 = 21 (ψ + kη) and k=
4ν(d − 1 − (d − 5)ν 2 ) . (d − 5)(d − 3)ν 4 − 2(d − 1)(d − 5)ν 2 + (d − 1)(d − 3)
(3.22)
The metric components are found to be 1 − ν 2 cos2 θ , 1 − µ2 cos2 θ 1 − ν 2 cos2 θ a1 = a2 = , ˜ 2 1 − µ2 cos2 θ 1 1 − ν2 sin2 θ, a3 = 4 1 − µ2 1 − ν 2 cos2 θ
(3.24)
b2 = cos2 θ,
(3.26)
h2 =
(3.23)
(3.25)
where µ2 =
(d − 3)ν 2 − (d − 5)ν 4 , d − 1 − (d − 3)ν 2
(3.27)
˜ is given by (3.17). and To avoid singularities, we will assume 0 ≤ ν 2 ≤ 1. In this range, µ2 and k are monotonously increasing functions with respect to ν. Then, we have 0 ≤ µ2 ≤ 1, −2 ≤ k ≤ 2 ˜ > 0 (d ≥ 4). By the analysis similar to Sect. 2, the ranges of angles must be and restricted as 0 ≤ θ ≤ π2 , 0 ≤ ψ ≤ 4π, 0 ≤ χ ≤ π and 0 ≤ η ≤ 2π . If we impose that k ∈ Z, the 1-form σ 3 can be regarded as a connection on the lens space Lk . Taking account of the inequality |k| ≤ 2, we have (i) |k| = 0,
(ii) |k| = 1,
(iii) |k| = 2.
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Case (i) corresponds to ν = 0, and then the metric (3.18) gives the product metric on S 2 × S d−2 ; 1 1 2 1 (σ ) + (σ 2 )2 + dθ 2 + sin2 θdψ 2 + cos2 θgS d−4 . (3.28) g0 = d −3 4 Case (iii) corresponds to ν = ±1, and then the metric (3.18) is the standard S d -metric after a modification of the angle ψ → 2ψ;
1 (3.29) g±1 = dθ 2 + sin2 θ (σ 1 )2 + (σ 2 )2 + (σ 3 )2 + cos2 θgS d−4 . 4 In case (ii), the 1-forms σ i (i = 1, 2, 3) are identified with the Maurer-Cartan forms of SU(2). Thus the metric is of cohomogeneity one with principal orbits S 3 × S d−4 . The orbits collapse to S 2 × S d−4 at θ = 0 and to S 3 × {point} at θ = π/2. Hence, the total space is the unit sphere bundle of the vector bundle H ⊕ Rd−3 over S 2 , where H is the Hopf bundle and Rd−3 is the trivial bundle of rank d − 3. Since the second Stiefel-Whitney class w2 (H ⊕ Rd−3 ) = w2 (H ) = 0 in H 2 (S 2 ; Z) = Z2 , the space is S d−2 , the non-trivial S d−2 -bundle over S 2 . S2× Theorem 3. Let ν be a real number satisfying ν 2 < 1 and (d − 5)(d
4ν(d − 1 − (d − 5)ν 2 ) = ±1. − 2(d − 1)(d − 5)ν 2 + (d − 1)(d − 3)
− 3)ν 4
(3.30)
S 2 , which is Then gν gives an Einstein metric with positive scalar curvature on S d−2 × d−2 2 the non-trivial S -bundle over S . Remark 6. For d = 4, this result reproduces the Page metric on CP 2 CP 2 . In this case the principal orbits are S 3 , the range of θ is extended to 0 ≤ θ ≤ π and the metric has Z2 -symmetry about θ = π/2. For d = 5, this represents the metric (2.50). Acknowledgements. We thank R. Goto, H. Ishihara and S. Tanimura for useful discussions. Y.Y. would like to express his gratitude to G.W. Gibbons and S.A. Hartnoll for useful discussions during his stay at DAMTP, Cambridge University. This paper is supported by the 21 COE program “Constitution of wideangle mathematical basis focused on knots”. Research of Y.H. is supported in part by the Grant-in Aid for scientific Research (No. 15540090) from Japan Ministry of Education. Research of Y.Y. is supported in part by the Grant-in Aid for scientific Research (No. 14540073 and No. 14540275) from Japan Ministry of Education.
References 1. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]; Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998); Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) 2. Hawking, S.W., Page, D.N.: Thermodynamics Of Black Holes In Anti-De Sitter Space. Commun. Math. Phys. 87, 577 (1983) 3. Witten, E.: Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv. Theor. Math. Phys. 2, 505 (1998) 4. Hawking, S.W., Hunter, C.J., Taylor-Robinson, M.M.: Rotation and the AdS/CFT correspondence. Phys. Rev. D 59, 064005 (1999) 5. Page, D.: A Compact Rotating Gravitational Instanton. Phys. Lett. B 79, 235 (1978) 6. B¨ohm, C.: Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math. 134, 145–176 (1998) 7. Wang, M.Y., Ziller, W.: Einstein metrics on principal torus bundles. J. Diff. Geom. 31, 215–248 (1990) Communicated by G.W. Gibbons
Commun. Math. Phys. 257, 287–290 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1261-x
Communications in
Mathematical Physics
Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map David Damanik1, Rowan Killip2 1 2
Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected] Department of Mathematics, University of California, Los Angeles, CA 90055, USA. E-mail: [email protected]
Received: 14 April 2004 / Accepted: 1 June 2004 Published online: 13 January 2005 – © Springer-Verlag 2005
Abstract: We show that discrete one-dimensional Schr¨odinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, Vθ (n) = f (2n θ), may be realized as the half-line restrictions of a non-deterministic family of whole-line operators. As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty. Consider discrete Schr¨odinger operators [Hθ φ](n) = φ(n + 1) + φ(n − 1) + λf (2n θ )φ(n)
(1)
on 2 (Z+ ), Z+ = {1, 2, . . . }, with a Dirichlet boundary condition, φ(0) = 0. Here, λ > 0 denotes the coupling constant, θ ∈ T = R/Z, and f : T → R is measurable, bounded, and non-constant. Since the doubling map, T : T → T, θ → 2θ mod 1, is ergodic with respect to Lebesgue measure on T, the Lyapunov exponent exists for every energy E, that is, there is a function γ : R → [0, ∞) such that 1 log M(n, E, θ ) for almost every θ, n→∞ n
γ (E) = lim
where M(n, E, θ ) is the transfer matrix from 1 to n for the operator Hθ at energy E ∈ R, that is, E − λf (2n θ ) −1 E − λf (2θ) −1 M(n, E, θ ) = × ··· × . 1 0 1 0 It is expected that γ (E) > 0 for every E and that Hθ has pure point spectrum with exponentially decaying eigenfunctions for almost every θ . The reason for this is that the potentials are strongly mixing and similar to random potentials (generated by i.i.d.
D. D. was supported in part by NSF grant DMS–0227289.
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random variables) for which these results were established in the 1980’s. There is a huge literature on this subject; we refer the reader to [4, 11] for detailed references. However, very few results are known for the operators Hθ and these are limited to the case of small coupling: Chulaevsky and Spencer, [5], proved an asymptotic formula for γ (E), in the limit λ → 0. This implied positivity of γ (E) for all 0 < |E| < 2 and λ small, which in turn was used by Bourgain and Schlag, [3], to prove that for λ < λ0 and almost every θ, Hθ has pure point spectrum in 0 < δ ≤ |E| ≤ 2 − δ with exponentially decaying eigenfunctions. This is usually referred to as Anderson localization. The results just described require a regularity assumption on f (e.g., H¨older continuity). Beyond small coupling, no spectral results have been established. In particular, it was not known, for any λ > 0, whether there can be any absolutely continuous spectrum (for a.e. θ ).1 In this note, we show that there is none. Theorem 1. Suppose that λ > 0 and f is measurable, bounded, and non-constant. Then, the Lyapunov exponent γ (E) is positive for almost every E ∈ R and the absolutely continuous spectrum of Hθ is empty for almost every θ ∈ T. Remark. Note that if the binary expansion of θ is periodic, the absolutely continuous spectrum of Hθ is not empty so that the second statement cannot be improved. As we mentioned above, it is expected that γ (E) is strictly positive for every energy E. Once this can be shown, the next natural step will then be to prove Anderson localization, using, for example, methods from [3]. Since such a result is currently out of reach, we note that, by general principles, the almost everywhere positivity of γ (E) yields the following consequence in terms of a localization result. Given α ∈ [0, π ), let Hθ,α denote the operator which acts on 2 (Z+ ) as in (1), but with φ(0) given by cos(α)φ(0) + sin(α)φ(1) = 0. Thus, the original operator (with a Dirichlet boundary condition) corresponds to α = 0. By [11, Theorem 13.4], Theorem 1 implies the following: Corollary 1. Suppose that λ > 0 and f is measurable, bounded, and non-constant. Then, for almost every α ∈ [0, π) and almost every θ ∈ T, the operator Hθ,α has pure point spectrum, and all eigenfunctions decay exponentially at infinity. It is interesting to note that there are two possible viewpoints and both have been used successfully. The work of Chulaevsky and Spencer showed that methods from the random case (specifically an approach developed by Figotin and Pastur [11]) could be extended to establish the asymptotic formula and positivity for the Lyapunov exponent. On the other hand, Anderson localization was then proven by Bourgain and Schlag by adapting methods originally developed for models with very little randomness, namely, with underlying dynamics given by the shift [1, 6, 7] and the skew-shift on the torus [2]. The main ingredient in the proof of Theorem 1 will be a result of Kotani, [9] (see [12] for an adaptation to the discrete case), which shows that for whole-line operators with non-deterministic potentials, the Lyapunov exponent is almost everywhere positive and, consequently, the absolutely continuous spectrum is almost surely empty. A potential is non-deterministic if it is not determined uniquely by its restriction to a half-line. Thus, the natural strategy will be to define a non-deterministic family of whole-line operators 1 It should be possible to exclude absolutely continuous spectrum using Kotani’s support theorem [10]; see [5, 13] for remarks indicating this possibility and [8] for related applications of the support theorem. However, this would not give absence of absolutely continuous spectrum in full generality, that is, this approach would only work for some f ’s and some values of λ.
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whose restrictions to the right half-line yield the family {Hθ }. Note that the Lyapunov exponents for the two families are the same and that the half-line operators cannot have absolutely continuous spectrum if the whole-line operators do not have any. Proof of Theorem 1. The first step is to conjugate the doubling map T to a symbolic Z+ shift ∞ via the−nbinary expansion. Let + = {0, 1} and define D : + → T by D(ω) = n=1 ωn 2 . The shift transformation, S : + → + , is given by (Sω)n = ωn+1 . Clearly, D ◦ S = T ◦ D. Next we introduce a family of whole-line operators as follows. Let = {0, 1}Z and define, for ω ∈ , the operator [Hω φ](n) = φ(n + 1) + φ(n − 1) + Vω (n)φ(n) in 2 (Z), where Vω (n) = f [D({ωn , ωn+1 , ωn+2 , . . . })]. The family {Hω }ω∈ is non-deterministic since Vω restricted to Z+ only depends on {ωn }n≥1 and hence, by non-constancy of f , we cannot determine the values of Vω (n) for n ≤ 0 uniquely from the knowledge of Vω (n) for n ≥ 1. It follows from [9, 12] that the Lyapunov exponent for {Hω }ω∈ is almost everywhere positive and σac (Hω ) is empty for almost every ω ∈ with respect to the ( 21 , 21 )-Bernoulli measure on . Finally, let us consider the restrictions of Hω to 2 (Z+ ), that is, let Hω+ = E ∗ Hω E, where E : 2 (Z+ ) → 2 (Z) is the natural embedding. Observe that Hω+ = Hθ , where θ = D({ω1 , ω2 , ω2 , . . . }). This immediately implies the statement on the positivity of the Lyapunov exponent for the family {Hθ }θ∈T . As finite-rank perturbations preserve absolutely continuous spectrum, σac (Hω+ ) ⊆ σac (Hω ) for every ω ∈ . This proves that σac (Hω+ ) = ∅ for almost every ω ∈ .
Remark. The proof relies on only two properties of the doubling map: it is not one-toone and it can be extended to a dynamical system over Z. Consequently, Theorem 1 extends to other models with these properties. For example, θ → mθ mod 1 for any integer m ≥ 2. Acknowledgments. We thank Svetlana Jitomirskaya and Barry Simon for useful conversations.
References 1. Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math. 152, 835–879 (2000) 2. Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schr¨odinger operators on Z with potentials given by the skew-shift. Commun. Math. Phys. 220, 583–621 (2001) 3. Bourgain, J., Schlag, W.: Anderson localization for Schr¨odinger operators on Z with strongly mixing potentials. Commun. Math. Phys. 215, 143–175 (2000) 4. Carmona, R., Lacroix, J.: Spectral Theory of Random Schr¨odinger Operators. Boston: Birkh¨auser, 1990 5. Chulaevsky, V., Spencer, T.: Positive Lyapunov exponents for a class of deterministic potentials. Commun. Math. Phys. 168, 455–466 (1995) 6. Goldstein, M., Schlag, W.: H¨older continuity of the integrated density of states for quasi-periodic Schr¨odinger equations and averages of shifts of subharmonic functions. Ann. of Math. 154, 155–203 (2001) 7. Jitomirskaya, S.: Metal-insulator transition for the almost Mathieu operator. Ann. of Math. 150, 1159–1175 (1999)
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8. Kirsch, W., Kotani, S., Simon, B.: Absence of absolutely continuous spectrum for some onedimensional random but deterministic Schr¨odinger operators. Ann. Inst. H. Poincar´e Phys. Th´eor. 42, 383–406 (1985) 9. Kotani, S.: Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schr¨odinger operators. In: Stochastic analysis (Katata/Kyoto, 1982), North-Holland Math. Library, 32, Amsterdam: North-Holland, 1984, pp. 225–247 10. Kotani, S.: Support theorems for random Schr¨odinger operators. Commun. Math. Phys. 97, 443–452 (1985) 11. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Grundlehren der Mathematischen Wissenschaften 297, Berlin: Springer-Verlag, 1992 12. Simon, B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983) 13. Spencer, T.: Ergodic Schr¨odinger operators. In: Analysis, et cetera, Boston: Academic Press, 1990, pp. 623–637 Communicated by B. Simon
Commun. Math. Phys. 257, 291–302 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1324-7
Communications in
Mathematical Physics
Self-Dual Strings in Six Dimensions: Anomalies, the ADE-Classification, and the World-Sheet W ZW -Model M˚ans Henningson Department of Theoretical Physics, Chalmers University of Technology and G¨oteborg University, 412 96 G¨oteborg, Sweden. E-mail: [email protected] Received: 12 May 2004 / Accepted: 12 November 2004 Published online: 22 March 2005 – © Springer-Verlag 2005
Abstract: We consider the (2, 0) supersymmetric theory of tensor multiplets and selfdual strings in six space-time dimensions. Space-time diffeomorphisms that leave the string world-sheet invariant appear as gauge transformations on the normal bundle of the world-sheet. The naive invariance of the model under such transformations is however explicitly broken by anomalies: The electromagnetic coupling of the string to the two-form gauge field of the tensor multiplet suffers from a classical anomaly, and there is also a one-loop quantum anomaly from the chiral fermions on the string world-sheet. Both of these contributions are proportional to the Euler class of the normal bundle of the string world-sheet, and consistency of the model requires that they cancel. This imposes strong constraints on possible models, which are found to obey an ADE-classification. We then consider the decoupled world-sheet theory that describes low-energy fluctuations (compared to the scale set by the string tension) around a configuration with a static, straight string. The anomaly structure determines this to be a supersymmetric version of the level one Wess-Zumino-Witten model based on the group (R × SU (2))/Z2 .
1. Introduction In a recent paper [1], we constructed a (2, 0) supersymmetric theory of tensor multiplets and self-dual strings in six dimensions. The model is formulated in terms of fields defined over a six-dimensional Minkowski space-time M, and fields defined over a three-dimensional Dirac-membrane world-volume D, the boundary ∂D of which equals the string world-sheet . The fields over M are scalars φ, chiral spinors ψ, and a twoform gauge-field b with gauge-invariant field-strength h = db. The fields over D are a Minkowski space vector X and a Minkowski space anti-chiral spinor . The (2, 0) supersymmetry algebra contains an SO(5) R-symmetry, under which φ transforms in the vector representation, ψ and transform in the spinor representation, and b and X are invariant. All fields obey certain reality conditions.
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This model is invariant under a local ‘κ-symmetry’, by means of which most of the fields X and may be gauged away. Although this was not explicitly shown in the paper, the remaining theory could then be described in terms of the fields φ, ψ, and b over M together with certain fields X ⊥ , + , and − defined over the string world-sheet . The latter can be understood as follows: The world-sheet field X defines an embedding X : → M. Consider the pullback ∗ (T M) = X ∗ (T M) of the tangent bundle T M of M to the string world-sheet by this map. (In this paper, a raised star ∗ will always denote pullbacks from M to by X.) This bundle splits as ∗ (T M) = T ⊕ N , where T is the tangent bundle of , and N is its orthogonal complement, i.e. the normal bundle of in M. Furthermore, the pullback ∗ (φ) of the scalar field φ to defines an embedding of in a five-dimensional real vector space, the normal bundle of which we denote as E. Associated to the SO(1, 1) bundle T , there are chiral spinor bundles + and − , where the superscript denotes the chirality. Similarly, associated to the SO(4) bundles N and E, there are chiral spinor bundles N + , N − and E + , E − respectively. All these bundles over are endowed with natural connections induced from the embedding of . The fields over the world-sheet are a bosonic section X ⊥ of N , describing transverse fluctuations of , together with fermionic sections + and − of + ⊗ N + ⊗ E + and − ⊗ N − ⊗ E − respectively. The model can be given a Lagrangian formulation with an action 1 ¯∂ψ + htot ∧ ∗htot VolM ∂φ∂φ + ψ/ S= 2 4π λ M ¯ +D ¯ −D + Vol ∗ (φφ) DX⊥ DX ⊥ + / ++ / − + e ∗ (b) + . . . , (1)
where the omitted terms will not be important in the present paper. Here VolM and Vol are the volume form on M and the induced volume form on respectively, ∗ denotes the Hodge duality operator, e is the electric charge of the string, and λ is a coupling constant. The world-sheet derivative D acting on sections of various bundles is constructed using the appropriate connection. The total field strength htot obeys a modified Bianchi identity dhtot = 2πq δ ,
(2)
where q is the magnetic charge of the string, and δ is the Poincar´e dual four-form of the string world-sheet . The anti self-dual part h− ≡ 21 (h + ∗h) of the field strength h, which is not part of the tensor multiplet, decouples from the rest of the theory provided that we choose the coupling constant so that λ2 = q/e. The group of diffeomorphisms of M is of course spontaneously broken by the string to the subgroup that leaves the string world-sheet invariant. From a world-sheet perspective, this subgroup appears as an SO(4) gauge group acting on the normal bundle N. Naively, the action (1) is invariant under such transformations. However, because of the modified Bianchi identity (2), the electric coupling term e ∗ (b) in fact transforms anomalously already at the classical level. 1 Furthermore, the chiral fermions + and − transform in complex representations of SO(4), so the transformation of the quantum effective action acquires further anomalous terms at the one-loop level. In the next section, we will describe these contributions to the total anomaly more carefully. 1 The construction in [1] involved regulating the theory by introducing a perturbation of the worldsheet . This is rather analogous to the framed knot discussed in [2]. As pointed out to me by E. Witten, the necessity to regulate the theory in this way is a symptom of the underlying anomaly.
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At this point, it should be stressed that we are considering a theory in flat six-dimensional Minkowski space, and we are not attempting to gauge the SO(5) R-symmetry. It is well known that in e.g. the (2, 0) supersymmetric world-volume theory on five-branes embedded in eleven-dimensional M-theory, the local diffeomorphism and SO(5) symmetries a priori suffer from anomalies. This would of course spoil the consistency of the theory. However, as shown in [3], thanks to delicate cancellations between the six-dimensional world-volume anomalies, anomaly inflow terms from the eleven-dimensional bulk, and a subtle contribution that originates from the Chern-Simons interaction of eleven-dimensional M-theory, the total anomalies in fact vanish. The six-dimensional theory is thus consistent when embedded into eleven-dimensional M-theory. However, it is not consistent in itself when coupled to an arbitrary curved background metric and SO(5) connection. But in the situation that we are considering here, i.e. a flat space and a global SO(5) R-symmetry, there is no problem, and we need not be concerned with gravitational and gauge anomalies. The particular string world-sheet anomalies that are the main subject of the present paper have, to the best of our knowledge, not been considered previously. (However, some related issues were discussed already in [4], and world-sheet anomalies on solitonic string solutions in a five-dimensional field theory are considered in [5].) Consistency of the model requires that the total anomaly cancels, and as we will see in Section Three, this imposes strong restrictions on possible six-dimensional (2, 0) theories. We will in fact find that they obey an ADE-classification, i.e. they are in oneto-one correspondence with the discrete subgroups of SU (2), or the simply laced Lie groups SU (r + 1), SO(2r), E6 , E7 , and E8 . This result was indeed predicted by the original definition of the (2, 0) theories in terms of ten-dimensional type IIB string theory on the product of M and a four-manifold with a simple singularity [6]. However, it is gratifying to recover it by a purely six-dimensional argument. The (2, 0) theories may also be regarded as the six-dimensional origin of certain N = 4 super Yang-Mills theories in four space-time dimensions, and from this point of view, it is of course more surprising that only models with a simply laced gauge group can appear. Because of its electromagnetic charge, a ‘bare’string is always surrounded by an electromagnetic field configuration, and as mentioned above, the classical anomaly arises from the strong interaction of the string with this self-field. In view of this, it is natural to try to define a ‘dressed’ string, which only interacts weakly with other excitations of the theory. A concrete situation to consider is a configuration with a static, straight string. The tension of the string is given by the vacuum expectation value of the scalar field φ. At energies low compared to the square root of this tension, the theory consists of a Minkowski space field theory (the free tensor multiplet theory) and a decoupled theory describing fluctuations of the dressed string. In Section Four, we will show that the latter world-sheet theory is a supersymmetric version of the level one Wess-Zumino-Witten model based on the group (R × SU (2))/Z2 . However, we would like to caution the reader that the results of this section are less well established than those in the preceding sections. 2. Classical and Quantum Anomalies 2.1. The descent formalism. On the two-dimensional world-sheet , an anomaly under gauge transformations that can be continuously connected to the identity (so called ‘perturbative anomalies’as opposed to ‘global anomalies’) is described by a four-dimensional integer characteristic class I . The two extra dimensions arise from the necessity of
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considering a two-parameter family of gauge-field configurations [7, 8]. The gauge invariant four-form I is closed. Locally, it can thus be written as I = dω for some threeform ω, which however will not be gauge invariant. Its variation under an infinitesimal gauge transformation is of the form δω = dA, where the two-form A is linear in the parameters of the transformation. The anomalous variation of the effective action is then given by 2π ∗ (A). 2.2. The electric coupling. To exhibit the anomaly of the electric coupling in (1), we can rewrite it in various ways, none of which is completely satisfactory, though, but should rather be seen as heuristic expressions: tot ∗ ∗ e (b) = e b ∧ δ = e h ∧ δD = e (htot ). (3)
M
M
D
Starting from the left, the first definition is given by integrating the pullback of b over the world-sheet . In the second version, we have used the Poincar´e dual δ of to rewrite it as an integral over space-time M. 2 The problem with these expressions is that they are not manifestly gauge-invariant, since they depend on the non-gauge-invariant quantity b. To remedy this, we can try to work solely with the gauge-invariant field strength htot as in the third expression. To incorporate the modified Bianchi identity (2) we define htot = db + 2πq δD , where δD is a three-form such that dδD = δ . Note that δD ∧δD = 0. But no matter how we choose δD , such a choice partly breaks the symmetry under diffeomorphisms of M that leave invariant. A particular choice, which is used in going to the last expression, is to take δD as the Poincar´e dual of an open three-manifold D, the boundary of which is given by . (This Poincar´e dual is defined in complete analogy with δ .) In this case, the unbroken subgroup consists of diffeomorphisms that leave D invariant. These difficulties are what we have in mind when we say that the electric coupling is anomalous. Under an infinitesimal gauge transformation, the variation of δD is of the form 1 δ(δD ) = qe dA, for some two-form A which is linear in the parameters of the transformation. The variation of the electric coupling, as given by the third expression in (3), is then e 1 tot tot ∗ h ∧ dA = dh ∧ A = 2π δ ∧ A = 2π (A), (4) qe M q M M where we have used the modified Bianchi identity (2). So this would seem to fit into the descent formalism, if we take the characteristic class I to equal qe ∗ (δ ). Indeed, this means that we can take ω = qe ∗ (δD ) so that δ(ω) = dA as required. As can be most clearly seen from the second expression in (3), this is in fact a particular case of anomaly inflow from the six-dimensional bulk to the two-dimensional world-sheet, with the only unfamiliar feature being that the anomaly four-form is given by the Poincar´e dual δ . 3 Since δ has delta-function support on , taking its pullback ∗ (δ ) might appear as a rather singular operation, but it actually has a well-defined meaning, as we will now The four-form δ is defined by the property that M δ ∧ s = ∗ (s) for an arbitary test-function two-form s. An explicit expression is δ = dx µ ∧ dx ν ∧ dx ρ ∧ dx σ dXτ ∧ dXκ δ (6) (x − X)µνρσ τ κ . 3 As discussed above, one should really consider a two-parameter family of world-sheets embedded in a two-parameter family of space-times. We will not make this explicit, though, but one should bear in mind that ∗ in these formulas denote the pullback to this two-parameter family of world-sheets rather than to . 2
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explain: In a tubular neighborhood of , we can approximate the space-time M with the total space of the normal bundle N . The Poincar´e dual δ then defines a class in the cohomology with compact vertical support Hv4 (N ) on this space. In fact, is the Thom class, i.e. the image of 1 under the Thom isomorphism H 0 () Hv4 (N ). We are thus interested in the pullback of by the zero section of N . But a general theorem states that this equals the Euler class χ (N ) of the normal bundle. (These matters are explained in more detail in textbooks on algebraic topology, see e.g. Sect. I.6 of [9] or Chap. 21 of [10].) So we find that the characteristic class I class associated with the classical anomaly of the electric coupling is given by I class = qe χ (N ).
(5)
2.3. The chiral fermions. As described in the introduction, the chiral fermions + and − are sections of + ⊗N + ⊗E + and − ⊗N − ⊗E − respectively. Here, ± , N ± , and E ± are the positive and negative chirality spinor bundles associated with the world-sheet tangent bundle T , the normal bundle N , and the R-symmetry bundle E respectively. The contributions to the anomalies now follow from standard formulas: For + and − we get ch2 (N + ) and −ch2 (N − ) respectively, where ch2 (N + ) and ch2 (N − ) denote the second Chern character classes. (A factor 21 due to the reality conditions on + and − cancels against a factor 2 corresponding to the rank of the bundles E + and E − .) The classes ch2 (N + ) and ch2 (N − ) are related to the Euler class χ (N ) and the first Pontryagin class p1 (N ): χ (N ) = ch2 (N − ) − ch2 (N + ), p1 (N ) = ch2 (N − ) + ch2 (N + ).
(6)
These relationships reflect the fact that SO(4) Spin(4)/Z2 (SU (2) × SU (2))/Z2 ,
(7)
where SO(4) is the structure group of N , and the two SU (2) factors are the structure groups of N + and N − respectively. Attempting to invert these relationships to express the integer classes ch2 (N+ ) and ch2 (N − ) in terms of χ (N ) and p1 (N ), we find that the mod 2 reduction of χ (N ) + p1 (N ) is the obstruction to lifting the SO(4) bundle N to a Spin(4) bundle. So we find that the characteristic class I quant associated with the one-loop quantum anomaly of the chiral fermions is given by I quant = −χ (N ).
(8)
3. The ADE-Classification 3.1. The coupling constant. Before discussing the possibilities for the anomalies found in the previous section to cancel, we will determine the correct value of the coupling constant λ, that appears in the action (1). In the absence of charged strings, the field 1 tot strength htot is closed, and we have normalized it so that 2π h is a representative of an integer class. The replacement λ → 1/λ then defines an equivalent theory. This is
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completely analogous to the S-duality of four-dimensional Maxwell theory, or the T duality of a compact boson in two dimensions. Only the self-dual part of htot is part of the tensor multiplet, though, and for an irrational value of λ2 , we do not know how to define the corresponding quantum theory. But for a rational value of λ2 , the Hilbert space of htot is a finite sum (with the number of terms depending on the value of λ2 ), where each term is a tensor product of two Hilbert spaces, pertaining to the self-dual and anti-self-dual parts respectively. For a single free tensor multiplet, i.e. the world-volume theory of a single five-brane in M-theory, it appears that the correct value to use is λ2 = 2, analogous to the ‘free fermion radius’ for a boson in two dimensions. Indeed, the additional topological data provided by the embedding into M-theory is then precisely what is needed to pick out the correct term in this sum [11]. The value λ2 = 2 can also be determined by the requirement that observables associated with different closed spatial surfaces commute with each other [12]. As we mentioned in the introduction, for this value of λ, the decoupling of the anti-self-dual part of the field strength requires that the electric and magnetic charges e and q of a string are related as e = 21 q. 3.2. Anomaly cancellation. Sofar, we have been discussing a theory with a single tensor multiplet and a single type of string. We now generalize this by introducing a tensor multiplet that takes its values in a real vector space W endowed with a positive definite inner product denoted as w · w for w, w ∈ W . (The positivity requirement of the inner product is necessary for the positivity of the Hamiltonian.) We also introduce a discrete subset Q ⊂ W of allowed magnetic string charges q. We will now determine all possible such sets Q of allowed magnetic charges. A first restriction is obtained by considering a single string with magnetic charge q ∈ Q, and thus electric charge e = 21 q ∈ W . From our results in the previous section, it follows that the total anomaly of such a string, taking both classical and quantum contributions into account, is given by the characteristic class I = I class + I quant = 21 (q · q − 2)χ (N ). All other terms in the action (1) are non-anomalous. Cancellation of the anomaly thus requires that q · q = 2,
(9)
for all q ∈ Q. 4
3.3. Dirac quantization. Another restriction on the spectrum of allowed charges follows from considering two different strings with electric and magnetic charges (e, q) and (e , q ) respectively. (We temporarily relax the relationship between electric and magnetic charges.) To begin with, we suppose that the electric charge of the first string and the magnetic charge of the second string vanish, i.e. e = 0 and q = 0. Because of the magnetic charge q of the first string, there is a non-trivial magnetic field htot in space-time. On the complement M ∗ = M − of the world-sheet of the first string, htot is closed and defines a cohomology class [htot ] = q, where is an element of H 3 (M ∗ , Z). Because of the electric charge e of the second string, the quantum ‘wave function’ of this system is not a complex function but rather a section of a complex line-bundle L over the configuration space M. (This is the space of all configurations in 4 This condition ensures that perturbative anomalies cancel. One should then continue and also investigate possible global anomalies, but we will not pursue this issue in the present paper.
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which the two strings do not intersect each other.) This line-bundle is completely characterized by its Chern class c1 (L), which we can specify by evaluating it on all possible two-cycles s ∈ H2 (M, Z). Such a two-cycle s defines a three-cycle S ∈ H3 (M ∗ , Z), and we have that s c1 (L) = e · q S . The generalization to arbitrary charges (e, q) and (e , q ) is that s c1 (L) = (e · q + e · q ) S . One should note the relative plus sign between the terms, as opposed to the minus sign familiar from the theory of dyonic particles in four dimensions. As explained in [13], this difference can be understood by a careful consideration of the topology of the configuration space calM. Heuristically, it is related to the fact that the wedge product of two-forms in four dimensions is symmetric, whereas the wedge product of three-forms in six dimensions is anti-symmetric. This sign has profound consequences, though: In four dimensions, we usually do not allow the world-lines of mutually non-local dyons to intersect. The world-line of a single dyon of course ’intersects’ itself, but this does not lead to any problems, since the minus sign ensures that dyons of the same kind are mutually local. In six dimensions, however, the plus sign means that a dyonic string is not mutually local with itself, and this can be seen as the origin of the classical bosonic anomaly discussed in the previous section. From the integrality of the class c1 (L), it now follows that the number e · q + e · q must be an integer. Reinstating the relationship between electric and magnetic charges, e.g. e = 21 q and e = 21 q , we thus find that q · q ∈ Z
(10)
for all q, q ∈ Q. 3.4. The ADE-classification. The conditions (9) and (10) on the elements of Q are precisely those that define the roots of a simply laced Lie algebra, i.e. the algebras Ar su(r + 1) for r = 1, 2, . . . , Dr so(2r) for r = 4, 5, . . . , and Er for r = 6, 7, 8. We have thus recovered the ADE-classification of consistent (2, 0) theories by a purely six-dimensional argument. This means that we should think of the tensor multiplet as taking its values in the weight space W of the corresponding simply laced Lie algebra. The set Q of allowed magnetic charges can then be defined as
Q = q ∈ r q · q = 2 , (11) tot where r is the root lattice of this algebra. tot The three-form field strength h is subject 1 to the restriction that its periods 2π h , where the integral is taken over a three-cycle in M, should be elements of the weight lattice w ⊂ W , i.e. the dual of the root lattice r ⊂ W . Locally, htot = db for some two-form b, subject to gauge transformations of the form 1 b → b + b. The parameter b is a closed two-form whose periods 2π b, where the integral is taken over a two-cycle in M, are elements of w . The theory should be 1 invariant under such transformations. However, the factor in the relationship e = 21 q 2∗ implies that the exponentiated electric coupling exp i e (b) will in general only be invariant up to a sign. The theory thus appears to suffer from a global anomaly under such gauge transformations. (There is no perturbative anomaly, since the coupling is invariant under transformations with an exact parameter b.) Hopefully, this anomaly is cancelled by a similar sign ambiguity in the definition of the path integral measure for the fermions + and − , but we will not attempt to show this in the present paper.
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3.5. The Ar -model. It is instructive to consider the Ar model in somewhat more detail. It can be realized as the world-volume theory of r + 1 parallel five-branes in M-theory, each of which supports a tensor multiplet. Membranes stretching from one five-brane to another appear as r(r + 1) different types of strings in six dimensions. Their magnetic charges with the respect to the tensor multiplets are given by vectors q with r + 1 entries of the form q = (0, . . . , 0, +1, 0, . . . , 0, −1, 0, . . . , 0).
(12)
Indeed the set Q of such charges fulfills the conditions (9) and (10). One linear combination of the r + 1 tensor multiplets, namely the sum, decouples from all the strings. This is analogous to the world-volume theory of r + 1 D3-branes in type IIB string theory, where the gauge group is SU (r + 1) rather than U (r + 1), because a central U (1) factor locally decouples from the rest of the theory. We should thus focus our attention on the r-dimensional linear space W orthogonal to the sum of the tensor multiplets. This can naturally be identified with the weight space of the Ar su(r + 1) Lie algebra, and Q is then indeed the set of roots of this algebra. For a single tensor multiplet, the periods of htot take their values in Z, so for r + 1 tensor multiplets, we instead get Zr+1 . But an element k = (k1 , . . . , kr+1 ) of the latter 1 group may be decomposed as k = s(1, . . . , 1) + w, where s is some multiple of r+1 , w and w belongs to the Ar weight lattice defined as w = (w1 , · · · , wr+1 ) ∈
1 1 Z × ··· × Z wi − wj ∈ Z, w1 + . . . + wr+1 = 0 . r +1 r +1
The root lattice r , i.e the dual of w , is then given by
r = (q1 , · · · , qr+1 ) ∈ Z × · · · × Z q1 + . . . + qr+1 = 0 .
(13)
(14)
4. The Decoupled World-Sheet Theory So far we have been describing the theory in terms of ‘bare’string, and certain space-time fields φ, ψ, and htot . But for some purposes, these are not the most convenient variables to use. The reason is, that in the presence of a string, a configuration with vanishing space-time fields is not possible. Indeed, the modified Bianchi identity (2) is an example of this phenomenon. We would therefore like to change variables, and describe the theory in terms of fluctuations around a configuration with a ‘dressed’ string, which includes this self-field. A concrete situation where this would be useful is a configuration containing a straight, static string. The string is characterized √ by its tension, which is given by the vacuum expectation value of the scalar field φφ. At energies low compared to the scale set by (the square root of) the tension, we expect that the theory factorizes into a space-time sector and a world-sheet sector, that are weakly coupled to each other. In the infra-red limit, we expect them to decouple completely. (It is more convenient to describe the decoupling limit by considering fluctuations of some fixed wavelength while taking the string tension to infinity.) The space-time sector is then of course the theory of a free tensor multiplet, and the world-sheet theory must be some two-dimensional conformal field theory.
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4.1. The spherical variables. To formulate this decoupled world-sheet theory, we start by considering the normal bundle N of the world-sheet embedded in space-time M. As described in the introduction, the world-sheet field X ⊥ is a section of this bundle. We may rewrite N as N (R × S)/Z2 ,
(15)
where R is a real line bundle with fiber R, and S is a three-dimensional sphere bundle with fiber S 3 SU (2). The non-trivial element of Z2 acts by multiplication with −1 on R and as the antipodal map on S 3 (i.e. by multiplication with the non-trivial element of the center of SU (2)). This of course corresponds to writing X ⊥ in terms of a radius r ∈ R, which is a section of R, and three angular variables g ∈ SU (2), that constitute a section of S. The restriction to positive r, which is customary in spherical coordinates, is replaced by the Z2 equivalence relation. In this formulation, the SO(4) (SU (2) × SU (2))/Z2 structure group of N acts as r→
r, g→
ugv −1 ,
(16)
for u, v ∈ SU (2). Its action on the fermionic fields + and − is
u+ , + → − →
v− .
(17)
The fermionic fields are also doublets under the unbroken SO(4) ⊂ SO(5)R symmetry and obey a reality condition, but we will suppress these structures from our notation. The covariant exterior derivative D acting on the various fields is thus Dr Dg D+ D−
= dr, ˜ = dg + Ag − g A, + = (d + A) , ˜ −, = (d + A)
(18)
where the SO(4) connection on N is expressed as a pair of SU (2) connections A and A˜ transforming as A → uAu−1 − duu−1 , ˜ −1 − dvv −1 . A˜ → v Av
(19)
As usual, the corresponding covariant field strengths are defined as F = dA + A ∧ A ˜ and F˜ = d A˜ + A˜ ∧ A. 4.2. The gauged Wess-Zumino term. We now wish to construct the world-sheet theory describing fluctuations around a configuration with a straight, static string of infinite tension. The action is in fact largely determined by the anomaly structure described in the previous sections, and must take the form d 2 σ L + SW Z . (20) S=
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Here L is some gauge invariant local Lagrangian density, and SW Z is a non-local gauged Wess-Zumino term. To define the latter, we must extend the domain of definition of the field g from to an open three-manifold D, the boundary of which equals . We then have 1 ˜ −1 A SW Z = Tr g −1 dg A˜ + dgg −1 A − g Ag 4π 1 + Tr g −1 dg ∧ g −1 dg ∧ g −1 dg , (21) 12π D where Tr denotes the trace in the fundamental representation of SU (2). Since the integrand of the second term equals 2π times the generator of H 3 (S 3 , Z), SW Z is a level ˜ and one gauged Wess-Zumino term. It is a well-defined functional mod 2π of A, A, the restriction of g to . The reason for including it in the action (20) is that it has the correct anomalous transformation properties under (16) and (19). Indeed, a short calculation shows that 1 SW Z → SW Z + Tr u−1 duA − v −1 dv A˜ mod 2π. (22) 4π For u and v infinitesimally close to the unit element of SU (2), the anomalous variation of SW Z follows by applying the descent procedure to the characteristic class I class = ch2 (F ) − ch2 (F˜ ) =
1 1 Tr(F ∧ F ) − Tr(F˜ ∧ F˜ ). 8π 2 8π 2
(23)
As discussed in Section Two, this coincides with the classical anomaly of the electric coupling for a string with magnetic charge q such that q · q = 2 and electric charge e = 21 q. 4.3. The local terms. It remains to determine the local Lagrangian density L in (20). But this follows from various symmetry requirements, notably the conformal invariance of the model. The universality class of the model is in fact governed by the gauged Wess-Zumino term SW Z described in the previous subsection. We find that L is a sum of separate kinetic terms for the fields r, g, + , and − . Up to a field redefinition, the radial field r is a free non-compact boson with Lagrangian density Lr = −
1 D+ rD− r. 4π
(24)
For the angular field g, we have a gauged non-linear sigma-model Lagrangian density Lg =
1 Tr g −1 D+ gg −1 D− g . 4π
(25)
Finally, the fermionic fields + are − are governed by the gauged Dirac Lagrangian density L =
i + † ( ) D+ + + (− )† D− − . 4π
(26)
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In these formulas, D+ and D− denote the covariant derivatives with respect to the world-sheet light-cone coordinates σ + and σ − . The last term gives a one-loop contribution I quant to the anomaly of the quantum effective action, as described in Sect. Two. The normalizations of Lr and L are conventional, but the coefficient in front of Lg is significant. One could imagine that its ‘bare’ value is very large, corresponding to the large tension of the string. This means that the sigma-model is weakly coupled. Let us now consider a configuration with A = A˜ = 0 as is appropriate for a static, straight string. As described in [14], the coupling constant of Lg will then flow to a non-trivial infra-red fixed point determined by the coefficient of the Wess-Zumino term SW Z . The critical value can be most easily described by giving the variation of the total action (20) under a general variation δg of the field g: 1 1 d 2 σ Tr g −1 δg∂+ (g −1 ∂− g) = d 2 σ Tr δgg −1 ∂− (∂+ gg −1 ) , δS = 4π 4π (27) i.e. the chiral currents J− = g −1 ∂− g and J+ = ∂+ gg −1 are separately conserved at the critical point.
4.4. World-sheet supersymmetry. A static straight string configuration breaks half of the supersymmetries of the six-dimensional (2, 0) supersymmetry algebra. The broken symmetries have infinitesimal parameters λ+ and λ− with the same quantum numbers as the Goldstino fields + and − . They act non-linearly on the fields: δr δg δ+ δ−
= 0, = 0, = λ+ , = λ− .
(28)
The unbroken symmetries have infinitesimal parameters η+ and η− with quantum numbers that differ from those of + and − in that the world-sheet chiralities are reversed. They act linearly on the fields: −ig −1 δg + Iδr = + (η+ )† + η+ (+ )† , δ+ = (g −1 ∂− g + iI∂− r)η+ , δ− = 0
(29)
for the η+ transformations, and −iδgg −1 + Iδr = − (η− )† + η− (− )† , δ+ = 0, δ− = (∂+ gg −1 + iI∂+ r)η−
(30)
for the η− transformations. Here I is the 2 × 2 unit matrix. A straightforward computation shows that the total action (20) is invariant under these transformations when A = A˜ = 0.
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Acknowledgements. This work was inspired by a visit to the Institute for Advanced Study in Princeton. I would like to thank the Institute for its hospitality, and Edward Witten for drawing my attention to the issue of anomalies. I am supported by a Research Fellowship from the Royal Swedish Academy of Sciences (KVA).
References 1. Arvidsson, P., Flink, E., Henningson, M.: The (2, 0) supersymmetric theory of tensor multiplets and self-dual strings in six dimensions. JHEP 0405, 048 (2004) 2. Witten, E.: Quantum field theory and the jones polynomial. Commun. Math. Phys. 121, 351 (1989) 3. Freed, D., Harvey, J.A., Minasian, R., Moore, G.W.: Gravitational anomaly cancellation for M-theory fivebranes. Adv. Theor. Math. Phys. 2, 601 (1998) 4. Brax, P., Mourad, J.: Open supermembranes coupled to M-theory five-branes. Phys. Lett. B 416, 295 (1998) 5. Boyarsky, A., Harvey, J.A., Ruchayskiy, O.: A toy model of the M5-brane: Anomalies of monopole strings in five dimensions. Annals Phys. 301, 1 (2002) 6. Witten, E.: Some comments on string dynamics. http://arxiv.org/list/hep-th/9507121, 1995 7. Atiyah, M.F., Singer, I.M.: Dirac operators coupled to vector potentials. Proc. Nat. Acad. Sci. 81, 2597 (1984) 8. Zumino, B.: In: Relativity, Groups, and Topology II, B.S. deWitt, R. Stora (eds.), Amsterdam: North Holland, 1984 9. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Berlin-Heidelberg-NewYork: Springer Verlag, 1982 10. Madsen, I., Tornehave, J.: From Calculus to Cohomology: de Rham cohomology and characteristic classes. Cambridge: Cambridge University Press, 1997 11. Witten, E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103 (1997) 12. Henningson, M.: The quantum Hilbert space of a chiral two-form in d = 5+1 dimensions. JHEP 0203, 021 (2002) 13. Deser, S., Gomberoff,A., Henneaux, M., Teitelboim, C.: p-brane dyons and electric-magnetic duality. Nucl. Phys. B 520, 179 (1998) 14. Witten, E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455 (1984) Communicated by N.A. Nekrasov
Commun. Math. Phys. 257, 303–317 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1264-7
Communications in
Mathematical Physics
Dynamic Scaling in Miscible Viscous Fingering G. Menon1 , F. Otto2 1
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: [email protected] 2 Institute for Applied Mathematics, University of Bonn, Wegelerstr. 10, Bonn 53115, Germany. E-mail: [email protected] Received: 17 May 2004 / Accepted: 4 July 2004 Published online: 12 February 2005 – © Springer-Verlag 2005
Abstract: We consider dynamic scaling in gravity driven miscible viscous fingering. We prove rigorous one-sided bounds on bulk transport and coarsening in regimes of physical interest. The analysis relies on comparison with solutions to one-dimensional conservation laws, and new scale-invariant estimates. Our bounds on the size of the mixing layer are of two kinds: a naive bound that is sharp in the absence of diffusion, and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion. The naive bound is simple and robust, but does not yield the experimental speed of transport. In a reduced model derived by Wooding [20], we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments. Wooding’s model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion.
1. Introduction We study pattern formation and mixing generated by the gravity driven instability of an interface between two fluids in a porous medium. We may distinguish three stages in the evolution of the flow: (a) an early stage governed by the linear instability, (b) an intermediate stage with scaling behavior, and (c) a late stage. The linear stability analysis is classical [2, 9, 18] and describes the evolution in stage (a) well. The late stage (c) may be quite different depending on competing physical effects such as molecular diffusion or surface tension. Saffman and Taylor’s discovery of a family of traveling wave solutions (fingers), parametrized by λ ∈ [0, 1], has led to extensive work on finger selection [18]. Much of this work has been sophisticated linear stability and singular perturbation analyses examining the role of surface tension in selecting a finger (see [1, 19] for reviews). This analysis is directly related to the asymptotic profile (stage (c)) observed experimentally by Saffman and Taylor. It also provides a formal understanding of the stability of the coherent fingers in stage (b) . More precisely, it is assumed that even when there
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are many competing fingers, these are locally described by the Saffman-Taylor solution, and one of these (λ = 1/2 typically) is selected by an additional physical mechanism. In most experiments there is a broad range of active modes and in view of the instability, one may expect the evolution in stage (b) to be unpredictable. Yet experimental and numerical work shows that despite the unpredictability of fine details, certain statistics (size of the mixing layer, finger width) satisfy robust scaling laws. Little is known analytically about this fully nonlinear and physically interesting regime. Our goal is to obtain rigorous results on dynamic scaling for the simplest nontrivial model problem. We simplify matters by considering the gravity driven transport of a dilute solute s by convection and diffusion (miscible fingering). Then one may assume that the mobility is uniform, and after suitable non-dimensionalization (see [20] for a derivation) we have the system ∂t s + u · ∇s = s, s ∈ [0, 1], ∇ · u = 0, u + ∇p = −sez .
(1) (2) (3)
The domain is x = (y, z) ∈ [0, L]n−1 × R, n = 2, 3. Equation (3) is Darcy’s law: the velocity is linearly proportional to the driving force which comprises a pressure gradient and buoyancy (−sez ). The Peclet number, L, is a measure of the strength of diffusion. It is the only external parameter. We are interested in scaling behavior that is independent of L and boundary effects, and in particular the behavior as L → ∞. For convenience we use periodic boundary conditions in y. We consider initial conditions that are small perturbations of the flat unstable stratification. Figure 1 shows four snapshots of the evolution. After an initial transient, the system develops a mixing zone with an intricate network of fingers on a mesoscopic scale. The details of fingering are sensitive to intial data, but there is a remarkable statistical regularity observed in physical [20] and numerical experiments [10]:
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a(t)
b(t)
Fig. 2. Caricature of a(t) and b(t)
(a) The end-to-end width of the mixing zone is independent of L for large L, and it is typically t. √ (b) The fingers broaden at the rate O( t). Two features of these scaling laws are astonishing on closer inspection. (a) Diffusive slowdown (or the missing factor of 2): The fastest exact solutions in the absence of diffusion (Saffman-Taylor fingers with λ = 0) have speed 1, and would give a mixing zone of size 2t (not t). In particular, all Saffman-Taylor solutions with λ ∈ [0, 1/2) cannot be selected by a vanishing diffusion limit. (b) Coarsening is limited by diffusion, but experiments and numerical simulations show √ it is primarily driven by the convective coalescence of nearby fingers. Thus, the t width of fingers is not based on transverse spreading by diffusion. A rigorous formulation of dynamic scaling involves a definition of vertical and horizontal length scales (denoted a(t) and b(t) respectively as in Fig. 2), followed by upper and lower bounds of the form 1 − o(1) ≤
a(t) ≤ 1, t
b(t) c ≤ √ ≤ C, t
t 1
(4)
for some constants C ≥ c > 0, under minimal assumptions on initial data. The estimates on a measure the size of the mixing zone, and the constant is crucial. The estimates on b are a statement about the rate of coarsening, and the constant is not as important. But in (5) below), evolves such generality, (4) is false: the unstable stratification s0 (defined in√ diffusively without fingering. Therefore, for this solution a(t) ∼ t, and there is no coarsening since there are no fingers.We may use continuity in initial conditions to then construct solutions that coarsen arbitrarily slowly. It is a subtle problem to precisely eliminate such “unphysical” initial data using assumptions of genericity or randomness.
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We sidestep this issue altogether, and focus on physically meaningful estimates that are simple, natural and robust. What we prove are upper bounds on the potential energy, mean perimeter, and mixing entropies that scale in the natural way with time. Though we obtain only one-sided estimates, these are robust and free of any ansatz on the structure of the flow. This perspective has been used profitably in a wide range of problems [4, 6, 12, 17], and is similar in spirit to the now classical work of Howarth [11]. 2. Statement of Results 2.1. Definition of bulk quanitities. Let Q denote the spatial domain x := (y, z) ∈ [0, L]n−1 × R := D × R, n = 2, 3. We consider periodic boundary conditions in y. The unstable stratification 0, z < 0 s0 (z) = (5) 1, z ≥ 0 will serve as the main reference configuration. We are interested in estimates independent of the length scale L. It is thus natural to consider the horizontal average of a scalar field f : Q → R 1 ¯ f (z) = f (y, z) dy, (6) |D| D and normalized integrals of the form 1 − f dx := f (y, z) dy dz = f¯ dz. R |D| D R The gravitational potential energy of s(t, x) is defined by E(t) = − (s0 (z) − s(t, x)) z dx = (s0 (z) − s¯ (t, z)) z dz. R
(7)
(8)
To be more precise, E is the negative of the gravitational energy. Observe that since s ∈ [0, 1] we have E ≥ 0, and E = 0 if and √ only if s = s0 . E is also a measure of mass transported, and we shall define a = 2 6E (the choice of constant is explained in Remark 1 below). In order to measure the width of fingers, we define the mean perimeter P (t) = − |∇s(t, x)| dx =
1
Hn−1 [s −1 (c)] dc.
(9)
0
The second inequality is the co-area formula ( [21, Thm 2.7.1]) and justifies the terminology mean perimeter. One effect of diffusion is to smooth sharp transitions and create “mushy zones” where 0 < s < 1. The size of these mixing zones can be measured by “mixing entropies” that vanish in the pure phases where s ∈ {0, 1}. We will work mainly with the entropies H (t) = − s(1 − s) dx, S(t) = − − (s log s + (1 − s) log(1 − s)) dx. (10)
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2.2. Uniform estimates on bulk quantities. The following estimates are independent of L and provide an upper bound on a(t) and a lower bound on b(t). Theorem 1. Let s(t, x) be a classical solution to (1)–(3), with energy E(t), mixing entropy H (t), and perimeter P (t). Then lim sup t→∞
E(t) 1 ≤ , 2 t 6
lim sup t→∞
1 H (t) ≤ , t 3
(11)
and lim sup t→∞
1 t2
t
P 2 (τ )dτ ≤
0
π . 9
(12)
Remark 1. In a loose sense, the energy estimate (11) bounds s(t, x) by comparison to the rarefaction wave (entropy solution) to the following Riemann problem: ∂t su − ∂z (su (1 − su )) = 0, More explicitly, su (t, z) = s ∗ (z/t), where 0 s ∗ (ξ ) = 1+ξ 2 1
su (0, z) = s0 (z).
ξ < −1, −1 ≤ ξ ≤ 1, ξ > 1.
(13)
(14)
Thus, the end-to-end size of the mixing zone is a(t) = 2t. The energy associated to the profile su (t, z) is E(t) = t 2 /6 = a 2 /24. A similar (and more general) comparison of s(t, x) with an entropy solution for a suitable Riemann problem appears in earlier work by one of the authors [16]. However, notice that the estimate a(t) ≤ 2t is twice the experimental result a(t) ≤ t. Here our interest is in understanding this unexpected gap. Remark 2. Estimate (12) is an integrated version of the (unproven) pointwise inequality √ 2πt . (15) P (t) ≤ 3 More precisely, the largest C and α in a scaling ansatz P (t) = Ct α compatible with (12) are the values in (15). The bound on α may be interpreted as a lower bound on the width of fingers as follows. If we assume the typical form of s is as shown in Fig. 2, we see that a P (t) ≈ − |∂y s| dx ≈ (16) N (z) dz = a N¯ = , b |z|≤a/2 where N(z) is the number of fingers per unit width on any horizontal level z = const, N¯ is the mean number of fingers, and b = 1/N¯ is the mean wavelength of fingers. The upper estimate (15) now yields, √ a(t) 3 t b(t) ≥ ≥√ (17) P (t) 2π if a(t) = t. It is in this weak (but also robust) sense, that (12) is an estimate on coarsening.
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2.3. Sharp pointwise estimates in a reduced model. The crux of the problem is the focusing mechanism of convection and the subtle role of diffusion in arresting singularity formation. This is manifested experimentally as diffusive slowdown. A similar phenomenon is seen experimentally and numerically in the Rayleigh-Taylor instability [5, 7] though this is harder to analyze. The scaling a(t) ∼ Ct (or a(t) ∼ Ct 2 for the Rayleigh-Taylor instability) is clear on physical grounds. However, deeper insight is needed to find the sharp constant (the terminal speed or acceleration in experiments). We have been unable to improve Theorem 1 for the system (1–3) or to formulate an appropriate result on singularity formation. However, the following reduced 2-d model derived by Wooding [20] is more tractable to analysis: ∂t s + u · ∇s = s, s ∈ [0, 1], ∇ · u = 0, u = (v, w), w = s¯ − s.
(18) (19) (20)
Equation (20) is formally obtained from Darcy’s law when the horizontal and vertical scales separate (a(t) b(t)). Equations (2) and (3) imply w = −∂y2 s. If the height of fingers is much greater than their width (a(t) b(t)), it is natural to assume |∂y2 w| |∂z2 w|, and formally we have ∂y2 w = −∂y2 s, which is integrated to yield (20) (see [20] for details). The proof of Theorem 1 extends to the reduced system, and we have as before Theorem 2. Let s(t, x) be a classical solution to (18)–(20), with energy E(t), mixing entropy H (t), and perimeter P (t). Then E(t) H (t) 1 t 2 1 π 1 lim sup 2 ≤ , lim sup P (τ )dτ ≤ . ≤ , lim sup 2 (21) t 6 t 3 t 9 t→∞ t→∞ t→∞ 0 Theorem 2 is completely analogous to Theorem 1, and suggests the mixing zone grows as a(t) = 2t. But this is false. Theorem 3. Let s(t, x) be a classical solution to (18)–(20) with continuous initial data s(0, x) : Q → [0, 1] such that lim max s(y, z) = 0,
z→−∞
y
lim min s(y, z) = 1.
(22)
lim min s(t, y, ct) = 1.
(23)
z→∞ y
Then for any c > 21 , lim max s(t, y, −ct) = 0,
t→∞
y
t→∞ y
Remark 3. The pointwise estimates (23) show that the mixing zone does not spread faster than a(t) = t under mild localization assumptions on the initial data (22). Numerical calculations suggest that this estimate is sharp [15, p.88]. The mean speed of the finger tips in Wooding’s experiments is 0.446, or a(t) = 0.892t [20, Eq.15]. Remark 4. The slowdown of the finger speed by a factor of 1/2 is reminescent of finger selection by surface tension [19], and it is natural to say, the Saffman-Taylor finger of width λ = 1/2 is selected by diffusion. However, we stress that Theorem 3 is free of any assumptions on the structure of the solutions except for the localization assumption (22).
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2.4. Connections with scalar conservation laws. The connections with the Riemann problem (13) may be clarified further for the reduced model (18)–(20). Let us first neglect the effect of diffusion and formally pass to the sharp interface limit s ∈ {0, 1} a.e. We drop s in (18), and substitute for w from (20), to obtain ∂t s + ∂y (vs) + ∂z ((s − s)s) = 0.
(24)
Equation (24) possesses a remarkable closure property. In the absence of diffusion, the pointwise constraint s ∈ {0, 1} a.e. is preserved. Thus, when we average in y we find ws = (s − s)s = s 2 − s 2 = s 2 − s,
(25)
since nonlinearity does commute with averaging if s ∈ {0, 1} a.e. Since we are considering small perturbations of the flat interface, it is natural to choose initial data s(0, z) = s0 (z). In this formal limit, the evolution of s is determined by the Riemann problem ∂t s − ∂z (s(1 − s)) = 0,
s(0, z) = s0 (z).
(26)
The entropy solution to this Riemann problem is the rarefaction wave in (14). But this is ruled out by Theorem 3. In fact, the proof of Theorem 3 suggests that the physically appropriate self-similar weak solution to (13) is s¯ (t, z) = s # (z/t) := s # (ξ ), where ξ < − 21 , 0 (27) s # (ξ ) = 21 − 21 ≤ ξ ≤ 21 , 1 ξ > 21 . The main heuristic idea behind the proof of Theorem 3 is that there is always a sharp gradient at the fingertips. This is made precise by comparing solutions of (18)–(20)) to viscous shocks of Burgers equation. Thus the physically appropriate solution to (14) consists of two “unphysical” shocks propagating outwards at speed 1/2 (unphysical meaning that the shocks fail to satisfy Lax’s entropy condition, [13, p.9]). 3. Proof of Bulk Estimates 3.1. Main lemmas. Theorem 1 is based on energy balance, control of gradients using mixing entropies, and an interpolation argument linking the mixing entropies and energy. We formalize these ideas in the following lemmas. Lemma 1 (Energy balance). Let s(t, x) be a classical solution to Eqs. (1)–(3) with energy E(t) and mixing entropy H (t). Then E˙ = s(1 − s) dz − H (t) − − |∇p|2 dx + 1. (28) R
Lemma 2 (Growth of mixing entropies). Let s(t, x) be a classical solution to (1)–(3) with mixing entropies H and S. Then H˙ = 2 − |∇s|2 dx,
S˙ = −
|∇s|2 dx. s(1 − s)
(29)
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Lemma 3 (Interpolation). Let s : R → [0, 1] be measurable and let E = R (s0 − s) z dz. Then 2E s(1 − s) dz ≤ , (30) 3 R 2E − (s log s + (1 − s) log(1 − s)) dz ≤ π . (31) 3 R 3.2. Proof of Theorem 2. We combine Lemma 1 and Lemma 3 to obtain, 2E E˙ ≤ + 1. s(1 − s) dz + 1 ≤ 3 R
(32)
This estimate may be integrated to yield (11). The details are as follows. Let e(t) solve 2e e˙ = + 1, e(0) = E(0). (33) 3 We may integrate (33) explicitly to obtain the solution √
2e(t) 2e(0) 2e(t)/3 + 1 t − − log √ = . 3 3 3 2e(0)/3 + 1
(34)
We claim that for every t ≥ 0, E(t) ≤ e(t).
(35)
Indeed, if ε > 0 let eε (t) be the solution to (33) with eε (0) = E(0) + ε. We combine (32) and (33) and integrate to obtain
2 t eε (t) − E(t) ≥ ε + eε (τ ) − E(τ ) dτ. 3 0 Let T = inf{t ≥ 0 : eε (t) < E(t)}. We claim that T = ∞. Since ε > 0 we have T > 0. If T is finite, then we have eε (T ) = E(T ), which implies the contradiction 0 ≥ ε > 0. This proves (35). To estimate H , we observe that s(1 − s) dz − H (t) = − (s(1 − s) − s(1 − s)) dx = − (s − s)2 dx ≥ 0. R
Thus, we apply Lemma 3 again to find 2E(t) H (t) ≤ s(1 − s) dz ≤ . 3 R
(36)
The estimate (11) now follows from (34), (35), and (36). To prove (12) we apply the Cauchy-Schwarz inequality and (29) to obtain,
1/2 P (t) = − |∇s| ≤ − s(1 − s) −
|∇s|2 s(1 − s)
1/2
˙ 1/2 . = H 1/2 (S)
(37)
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We integrate in time to obtain
t
t
P (τ )dτ ≤ 2
0
0
˙ )dτ ≤ H (t)S(t) ≤ 2π E(t). H (τ )S(τ 3
(38)
In the second inequality we have used the monotonicity of H and S. In the third inequality we used (30) and (31). We combine (38) and (11) to obtain (12). This completes the proof of Theorem 1.
3.3. Proof of Lemma 1. Lemma 1 is a statement of energy balance. For any scalar field s˜ : R → R (˜s = s˜ (z)) such that s − s˜ ∈ L2 (Q) the elliptic system ∇ · u = 0,
u + ∇ p˜ = (˜s − s)ez
is a Helmholtz decomposition of the vector field (˜s −s)ez , and we have the orthogonality relations − |u|2 dx + − |∇ p| ˜ dx = − (s − s˜ )2 dx, − u · ∇ p˜ dx = 0. (39) Observe that there is no convection unless s oscillates in y: if s(y, z) = s(z), then u = 0. The velocity u is uniquely determined by s, but p˜ depends on the background field s˜ . We choose s˜ = s to obtain 2 2 − |u| dx = − (s − s) dx − − |∇p|2 dx = s(1 − s)dz − − s(1 − s) dx − − |∇p|2 dx. (40) R
We substitute (8) in (1), integrate by parts, and use (3), (39) and the boundary conditions to find ˙ E = − − sez · u dx + − ∇s · ez dx = − |u|2 dx + 1. (41) Lemma 1 follows from (40) and (41).
3.4. Proof of Lemma 2. Lemma 2 is a particular consequence of the growth of concave entropies. Let g : [0, 1] → [0, ∞) be a smooth concave function such that g(0) = g(1) = 0. Let s(t, x) be a classical solution to (1). We multiply Eq. (1) by g (s) and integrate to obtain d − g(s(t, x)) dx = − − ∇ · (g(s)u) dx − − g (s)∇ · ∇s dx = − − g
(s)|∇s|2 dx, dt after integration by parts.
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3.5. Proof of Lemma 3. Lemma 3 is a corollary of the following general scale-invariant interpolation inequality. Theorem 4. Assume g : [0, 1] → [0, ∞) is a concave, symmetric (that is g(s) = g(1 − s)) entropy that satisfies the growth condition g(s) ≤ Cα s α ,
for some α >
1 . 2
(42)
Then if s : R → [0, 1] is measurable we have
1/2 g(s(z)) dz ≤ Cg (s0 − s)z dz = Cg E 1/2 . R
R
(43)
The sharp constant Cg is given by Cg = 2
1
g (s)2 ds
1/2 (44)
.
0
The inequality is strict unless s(z) = sg (z/t) for some t > 0, where sg (ξ ) is the optimal profile defined implicitly by g (sg (ξ )) = ξ,
ξ ∈ R.
(45) √
Remark 5. A growth condition such as (42) is necessary. If g = s(1 − s) we may consider a profile such that |s − s0 | = (|z| log |z|)−2 for large z. Then E is finite, but R g(s)dz is not. Remark 6. The optimal profiles in (45) are the rarefaction waves (entropy solutions) to the following Riemann problem: ∂t s − ∂z (g(s)) = 0, s(0, z) = s0 (z). √ If g = s(1 − s), then Cg = 2/3 and the optimal profile√is the linear rarefaction wave in (14). If g = − (s log s + (1 − s) log(1 − s)), Cg = π 2/3. Proof. 1. Symmetrization. Given s : R → [0, 1] define its symmetrization ssymm (z) =
1 (s(z) + 1 − s(−z)) . 2
(46)
Observe that ssymm is symmetric about the origin in the sense that ssymm (z) = 1 − ssymm (−z). E is unchanged under symmetrization, that is (s0 − ssymm ) z dz = (s0 − s) z dz. R
R
On the other hand, since g is concave and symmetric we have g(ssymm (z)) ≥
1 1 (g(s(z)) + g(1 − s(−z))) = (g(s(z)) + g(s(−z))) . 2 2
(47)
(48)
Dynamic Scaling in Miscible Viscous Fingering
Therefore,
313
R
g(ssymm (z)) dz ≥
R
(49)
g(s(z)) dz.
2. Rearrangement. We now consider the increasing rearrangement srearr of ssymm . Rearrangement does not change the distribution function of ssymm ( [14, Ch.3]) and we have g(srearr (z)) dz = g(ssymm (z)) dz. (50) R
R
On the other, rearrangement decreases the potential energy. This is easily seen when ssymm is a simple function, and the general case follows by approximation. 3. Henceforth, we will suppose that s(z) = srearr (z). We will first show that there is some constant C such that R g(s)ds ≤ CE 1/2 and then find the sharp constant and optimal profile. In the following, C denotes a constant that depends only on α and g that may increase from line to line. By the symmetry of s and g it suffices to consider 0 −∞ g(s(z))dz. Let θ > 0. We then have
0 −∞
g(s(z))dz =
0 −θ
g(s(z))dz +
≤ θ g∞ + C
−θ −∞
≤ θg∞ + C
−θ −∞
g(s(z))dz
s α (z)dz
0
α |z|s(z)dz
−∞ α 1−2α
≤ θ g∞ + CE θ
−θ −∞
|z|
−α/(1−α)
1−α dz (51)
.
−1/2α
We optimize and substitute θ = g∞ E 1/2 in (51) to obtain 1−1/2α 1/2 g(s(z))dz ≤ Cg∞ E . R
4. The best profile and constant: The sharp constant is g(s(z)) dz Cg = sup R 1/2 , E s
(52)
(53)
where the supremum is taken over all s : R → [0, 1] measurable. As we have seen, we may restrict attention to increasing, symmetric s. In this case, we may identify s as a probability distribution function, and consider Lebesgue-Stieltjes integrals with respect to the positive measure s(dz) [8]. We will now consider the right inverse of s(z) written as z(s). Then we have 2 z 1 1 E = (s0 − s(z)) z dz = (z(s))2 ds. (54) s(dz) = 2 0 R R 2 Moreover, we may also write g(s(z)) dz = R
1
g(s) 0
dz ds = − ds
0
1
g (s)z(s) ds.
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It follows from the Cauchy-Schwarz inequality and (54) that
R
1
g(s(z)) dz ≤
1/2
1/2
1
2
g (s) ds
(z(s)) ds
0
1
= 2E
2
0
1/2 2
g (s) ds
.
0
The inequality is sharp if and only if z(s) = tg (s) for some t > 0.
4. Diffusive Slowdown 4.1. Bulk estimates and diffusion. The upper estimate a(t) ≤ 2t in Theorem 1 does not account for the effect of diffusion. The same estimate is obtained if we neglect diffusion, and rewrite Eq. (1) as ∂t s + u · ∇s = 0. Mass is now transported only by convection, and (41) changes to E˙ = −|u|2 . We now use (40) to obtain E˙ =
R
s(1 − s) dz − H −
|∇p| ≤ 2
2E , 3
(55)
which we integrate to obtain E(t) ≤ t 2 /6 as earlier. Moreover, this naive upper bound is sharp if we consider a weak solution obtained as the limit of a periodic array of SaffmanTaylor fingers. A similar analysis on the reduced model (18)–(19) yields the analogous (and simpler) estimate 2E 2 . s(1 − s)dz − H ≤ E˙ = − (s − s) dx = 3 R
(56)
One effect of diffusion is to produce molecularly mixed “mushy zones” where 0 < s < 1. If these zones are sufficiently large, then they act as drags on the bulk motion. More precisely, the existence of lower bounds of the form H (t) ≥ c, lim inf t→∞ t
or
lim inf t→∞
|∇s|2 dx ≥ c,
(57)
for some c > 0, coupled with (28) shows that lim supt→∞ 6E/t 2 < 1 (strict inequality). However, neither inequality in (57) is true in full generality (initial data s0 serves as a counterexample again). We have been unable so far to prove diffusive slow down in (1)– (2) by this argument. It is worth noting that obtaining similar bounds is a key obstruction in mathematical studies of turbulence [3, Sect. 3]. Nevertheless, the estimates in (57) provide a valuable heuristic hint about the role of gradients and diffusion.
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4.2. Proof of Theorem 3. We construct upper and lower solutions that bound the spreading of solutions to (18)–(20). The main heuristic idea is the gradients are always sharp at fingertips. This suggests comparing s(t, x) with a suitable viscous shock profile. By the symmetry of the problem, it suffices to bound the downward spreading by an upper solution. The upper solutions are viscous shock profiles for Burgers equation (more precisely, Burgers equation with a concave flux −s 2 /2), that is 2
s ∂t s∗ − ∂z ∗ = ∂z2 s∗ . (58) 2 We consider viscous shocks that connect the states ε > 0 and 1 + ε at ∓∞ respectively. ε > 0 may be chosen arbitrarily small. The viscous shock profiles are found by making the traveling wave ansatz s∗ (t, x) = sε (z + cε t) := sε (ζ ) in (58). The only admissible speed cε is determined by the Rankine-Hugoniot condition, cε =
1 1 (1 + ε)2 − ε 2 = + ε. 2 1+ε−ε 2
(59)
The shock profiles solve the differential equation dsε 1 = (1 + ε − sε ) (sε − ε) . dζ 2 Thus, sε is strictly increasing and given explicitly by
1 ζ − z0 sε (ζ ) = ε + 1 + tanh , 2 4
(60)
(61)
where z0 is an arbitrary constant that reflects the invariance of (58) under translations in z. In order to find lower solutions, we transform (58) under the symmetry s∗ → 1 − s∗ , z → −z, to obtain,
(1 − s˜∗ )2 ∂t (1 − s˜∗ ) + ∂z = ∂z2 (1 − s˜∗ ). (62) 2 The viscous shock profile that connects the states ε, 1+ε at ∓∞ is s˜∗ (t, x) = sε (z−cε t). The speed cε and profile sε are given by (59) and (61) respectively. Theorem 3 now follows from the following lemma. Lemma 4. Assume s(t, x) is a classical solution to (18)–(19) with continuous initial data s(0, x). (a) If s(0, x) < s∗ (0, x), then s(t, x) < s∗ (t, x) for all t ≥ 0. (b) Similarly, if s(0, x) > s˜∗ (0, x), then s(t, x) > s˜∗ (t, x) for all t ≥ 0. Proof (of Theorem 3). Fix c > 1/2. Let ε be arbitrary with
1 1 0<ε≤ c− . 2 2 Then by (59) c − cε ≥ ε > 0.
(63)
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Since limz→−∞ maxy s(0, y, z) = 0, we may choose z0 in (61) such that s(0, x) < s∗ (0, x) for all x. By Lemma 4 we then have s(t, y, −ct) < s∗ (t, −ct) = sε ((cε − c)t). In view of (61) and (63), this yields lim sup max s(t, y, −ct) ≤ ε. y
t→∞
Since ε was arbitrary, we obtain as desired lim max s(t, y, −ct) = 0.
t→∞
y
The proof of the lower estimate in (23) is similar, and is omitted.
Proof (of Lemma 4). The proof is a direct application of the maximum principle. We write (18) in non-divergence form ∂t s + v∂y s + (s − s)∂z s − s = 0,
(64)
and compare it with (58) rewritten as ∂t s∗ + v∂y s∗ + (s − s∗ )∂z s∗ − s∗ = s∂z s∗ .
(65)
Let θ = s∗ − s. We subtract (64) from (65), and rearrange terms to obtain ∂t θ + v∂y θ + w∂z θ − θ ∂z s∗ − θ = s∂z s∗ .
(66)
We notice that by the strong maximum principle for (18) we have s > 0 for t > 0 and thus also s > 0 for t > 0. On the other hand, ∂z s∗ > 0 as can be seen from (61). Hence the r. h. s. of (66) is strictly positive s∂z s∗ > 0
for t > 0.
(67)
We now argue by the maximum principle. Assume θ ≥ 0 was not true. Since θ(0, x) ≥ 0 and limz→±∞ θ (t, y, z) = ε uniformly in (t, y), there exists a (t∗ , x∗ ) ∈ (0, ∞) × R2 such that θ (t∗ , x∗ ) = 0
and θ (t, x) ≥ 0 ∀(t, x) ∈ (0, t∗ ) × R2 .
In particular, ∂t θ (t∗ , x∗ ) = ∂y θ (t∗ , x∗ ) = ∂z θ (t∗ , x∗ ) = 0
and
θ(t∗ , x∗ ) ≥ 0.
(68)
Hence by (66) we would obtain s∂z s∗ (t∗ , x∗ ) ≤ 0 — in contradiction to (67). The proof of the lower estimate is identical. Redefine θ = s − s˜∗ . We then have ∂t θ + v∂y θ + w∂z θ − θ ∂z s˜∗ − θ = (1 − s)∂z s˜∗ , and (68) holds again at a point of minimum.
(69)
Acknowledgement. This work was supported by NSF DMS 03-05985, SFB 611 of the German Science Foundation at the University of Bonn, and the Max Planck Institute for Mathematics in the Sciences, Leipzig. G.M. acknowledges with pleasure the hospitality of the University of Bonn, and the MPI, Leipzig. F.O. acknowledges partial support through SFB 611.
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References 1. Batchelor, G.K., Moffatt, H.K., Worster, M.G., eds.: Perspectives in fluid dynamics. Cambridge: Cambridge University Press, 2000 2. Chouke, R., van Meurs P., van der Poel, C.: The instability of slow, immiscible, viscous liquid-liquid displacements. Trans. AIME 216, 188–194 (1958) 3. Constantin, P.: Some open problems and research directions in the mathematical study of fluid dynamics. In: Mathematics Unlimited—2001 and beyond, Berlin: Springer, 2001, pp. 353–360 4. Constantin, P., Kiselev, A., Oberman, A., Ryzhik, L.: Bulk burning rate in passive-reactive diffusion. Arch. Ration. Mech. Anal. 154, 53–91 (2000) 5. Dimonte, G.: Nonlinear evolution of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Physics of Plasmas, 6, 2009–2015 (1999) 6. Doering, C.R., Constantin, P.: Bounds for heat transport in a porous layer. J. Fluid Mech. 376, 263–296 (1998) 7. George, E., Glimm, J., Li, X.-L., Marchese, A., Xu, Z.-L.: A comparison of experimental, theoretical and numerical simulation of Rayleigh-Taylor mixing rates. PNAS 99, 2587–2592 (2002) 8. Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition, New York: John Wiley & Sons Inc., 1971 9. Hill, S.: Channelling in packed columns. Chem. Eng. Sci. 1, 247–253 (1952) 10. Homsy, G.M.: Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271–311 (1987) 11. Howarth, L.N.: Bounds on flow quantities. Ann. Rev. Fluid. Mech. 4, 1972 12. Kohn, R.V., Yan, X.: Upper bound on the coarsening rate for an epitaxial growth model. Comm. Pure Appl. Math. 56, 1549–1564 (2003) 13. Lax, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Philadelphia, Pa.: Society for Industrial and Applied Mathematics, 1973 14. Lieb, E.H., Loss, M.: Analysis. Providence, RI: American Mathematical Society, 1997 15. Manickam, O., Homsy, G.M.: Fingering instabilities in vertical displacement flows in porous media. J. Fluid. Mech. 288, 75–102 (1995) 16. Otto, F.: Evolution of microstructure: an example. In: Ergodic theory, analysis, and efficient simulation of dynamical systems, Berlin: Springer, 2001, pp. 501–522 17. Otto, F.: Cross-over in scaling laws: a simple example from micromagnetics. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Beijing: Higher Ed. Press, 2002, pp. 829–838 18. Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. London. Ser. A 245, 312–329 (1958) (2 plates) 19. Tanveer, S.: Surprises in viscous fingering. J. Fluid. Mech. 428, 511–545 (2000) 20. Wooding, R.A.: Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid. Mech. 39, 477–495 (1969) 21. Ziemer, W.P.: Weakly differentiable functions. Berlin-Heidelberg-NewYork: Springer-Verlag, 1989 Communicated by P. Constantin
Commun. Math. Phys. 257, 319–362 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1325-6
Communications in
Mathematical Physics
Degenerate Elliptic Resonances Guido Gentile1 , Giovanni Gallavotti2 1 2
Dipartimento di Matematica, Universit`a di Roma Tre, 00146 Roma, Italy Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, 00185 Roma, Italy
Received: 21 May 2004 / Accepted: 21 October 2004 Published online: 30 March 2005 – © Springer-Verlag 2005
Abstract: Quasi-periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations. The parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations. Here we prove that, for a class of elliptic tori, a resummation algorithm can be devised and proved to be convergent, thus extending to such lower-dimensional invariant tori the methods employed to prove convergence of the Lindstedt series either for the maximal (i.e. KAM) tori or for the hyperbolic lower-dimensional invariant tori. 1. Introduction Quasi-integrable analytic Hamiltonian systems are described by Hamiltonians of the form H = H0 (I) + εH1 (ϕ, I), where (ϕ, I) ∈ Td × A, with A an open domain in Rd , are conjugate coordinates (called angle-action variables), the functions H0 and H1 are analytic in their arguments, and ε is a small real parameter. We shall consider for simplicity only Hamiltonians of the form H=
1 I · I + εf (ϕ), 2
(1.1)
where · denotes the inner product in Rd . Kolmogorov’s theorem (KAM theorem) yields, for ε small enough, the existence of many invariant tori for Hamiltonian systems of the form (1.1): such tori can be parameterized by the corresponding rotation vectors, at least if the latter satisfy some Diophantine conditions. On the other hand Poincar´e’s theorem states the existence of periodic orbits, which can be parameterized by rotation vectors satisfying d − 1 resonance conditions (so that after a simple linear canonical map one can assume that the rotation vector is (ω1 , 0, 0, . . . , 0)).
320
G. Gentile, G. Gallavotti
A natural question is what happens for the invariant tori corresponding, in absence of perturbations, to rotation vectors satisfying s resonance conditions, with 1 ≤ s ≤ d − 2. If we fix the rotation vector as (ω, 0) ≡ (ω1 , . . . , ωr , 0, . . . , 0) and parameterize the invariant torus for ε = 0 with the action value I = 0 then, after translating the origin in Rd by (ω, 0) and setting I = (A, B) ∈ Rr × Rs , ϕ = (α, β) ∈ Tr × Ts , the Hamiltonian (1.1) becomes 1 1 H = ω · A + A · A + B · B + εf (α, β), 2 2
(1.2)
where (α, A) ∈ Tr × Rr and (β, B) ∈ Ts × Rs are conjugate variables, with r + s = d, and · denotes the inner product both in Rr and in Rs . Here we impose that ω is a vector in Rr satisfying |ω · ν| ≥ C0 |ν|−τ0
∀ ν ∈ Zr \ {0},
(1.3)
with C0 > 0 and τ0 ≥ r − 1, which is called the Diophantine condition; we shall define by Dτ0 (C0 ) the set of rotation vectors in Rr satisfying (1.3). We also write eiν·α fν (β). (1.4) f (α, β) = ν∈Zr
We shall suppose that f is analytic in a strip around the real axis of the variables α, β, so q q that there exist constants F0 , F1 , κ0 such that |∂β fν (β)| ≤ q!F0 F1 e−κ0 |ν| for all ν ∈ Zr s and all β ∈ T . There are quite a few results on the above problem; we summarize our understanding of the existing results in Appendix A1. Lower-dimensional tori are in general considered for pertubations of systems consisting in a collection of rotators and of oscillators. The frequencies of the rotators are called proper or basic frequencies, while the frequencies of the oscillators are called normal frequencies. The model we study corresponds to the case in which the normal frequencies vanish for ε = 0, and become of order ε as an effect of the perturbation. This is the reason why we speak of degenerate lower-dimensional tori. Such tori will be called elliptic or hyperbolic or mixed according to the signs of the normal frequencies to order ε, which are positive or negative or of mixed signs, respectively. The case of frequencies vanishing also to order ε (and possibly to any fixed order in ε) is of course interesting, and it has not yet been solved in complete generality. Partial results (for the case of only one normal frequency) have been obtained in Refs. [Ch1, Ch2]. The equations of motion for the system (1.2), written in terms of the angle variables alone, are α¨ = −ε∂α f (α, β),
β¨ = −ε∂β f (α, β),
(1.5)
so that, once a solution of (1.5) is found, the action variables are immediately obtained ˙ by a simple differentiation: A = α˙ − ω, B = β. We look for solutions of (1.5), for ε = 0, conjugated to the free solution (α 0 + ωt, β 0 , 0, 0), i.e. we look for solutions of the form α(t) = ψ + a(ψ, β 0 ; ε),
β(t) = β 0 + b(ψ, β 0 ; ε),
(1.6)
for some functions a and b, real analytic and 2π-periodic in ψ ∈ Tr , such that the motion in the variable ψ is governed by the equation ψ˙ = ω. We shall prove the following result.
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Theorem 1. Consider the Hamiltonian (1.2), with ω ∈ Dτ0 (C0 ) and f analytic and periodic in both variables. Suppose β 0 to be such that ∂β f0 (β 0 ) = 0,
(1.7)
and assume that the eigenvalues a1 , . . . , as of the matrix ∂β2 f0 (β 0 ) are pairwise distinct and strictly positive, i.e. for some constant a > 0 one has ai , aj − ai > a > 0 for all j > i = 1, . . . , s. Then there exist a constant ε > 0 and a set E ⊂ (0, ε) such that the following holds: (i) For all ε ∈ E there are solutions of (1.5) of the form (1.6), where the two functions a(ψ, β 0 ; ε) and b(ψ, β 0 ; ε) are real analytic and 2π -periodic in the variables ψ ∈ Tr . (ii) The relative Lebesgue measure of E ∩ (0, ε) with respect to (0, ε) tends to 1 as ε → 0. (iii) The functions a, b can be extended to Lipschitz functions of ε, ψ in [0, ε] × Tr . Remarks. (1) If the equations are linearized around the torus one realizes that the square roots of the eigenvalues of ε∂β2 f0 (β 0 ) are the frequencies controlling the motion near the unperturbed torus α = ψ, β = β 0 . In a linear approximation the negative eigenvalues correspond to exponential instability of the B coordinates, while the positive ones correspond to oscillatory instability, hence they are called hyperbolic and elliptic frequencies, respectively; if they are all negative the torus is hyperbolic and if they are all positive it is elliptic. From the literature one might expect that the non-resonance condition on the eigenvalues of ∂β2 f0 (β 0 ) could be avoided; see Appendix A1. (2) The case of negative ε was dealt with in Ref. [GG], with techniques close to the ones introduced here, and it corresponds to the case of hyperbolic tori. (3) The case of mixed stationarity, i.e. det ∂β2 f0 (β 0 ) = 0 and eigenvalues of ∂β2 f0 (β 0 ) of mixed signs (with non-degeneracy of the positive ones), can be treated in exactly the same way discussed in this paper and the above result extends to this case; cf. Theorem 2 in Sect. 7. (4) For ε ∈ E the smooth extension in (iii) does not represent parametric equations of invariant tori: it just says that their values in the physically interesting set E (which turns out to have dense complement in [0, ε]) can be smoothly interpolated in ε. Such (non-unique) extensions are commonly used for interpolation purposes and are called Whitney extensions. The novelty and the purpose of our work is the development of a method of proof based on the existence of a formal power series expansion for the functions (a, b) and its multiscale analysis producing a rearrangement of its terms, involving summing many divergent series, which turns it into an absolutely convergent series. The paper is organized as follows. In Sect. 2 we recall the basic formalism, following Ref. [GG], and in Sect. 3 we give a simple example of resummation. In Sect. 4 we set up terminology and discuss heuristically the ideas governing our resummations, by explaining why they have to be performed by a multiscale analysis of the series (which we call Lindstedt series) representing a formal expansion of the quasi-periodic motions in powers of ε. The singularities are first “probed” down to a scale in which possible resonances between the proper frequencies, i.e. the components of ω, and the normal frequencies, i.e. the square roots of the eigenvalues of ε∂β2 f0 (β 0 ), are still irrelevant. The analysis of such singularities leads to what we call non-resonant or high frequency resummations, which can be treated by the method of Ref. [GG], i.e. of the hyperbolic case, in which
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no resonances at all were possible between proper frequencies and normal frequencies (simply because, for the Hamiltonian (1.2) the latter did not exist). Further probing of the singularities leads to what we call the resonant (or infrared) resummations: the analysis is more elaborated and it requires new ideas, obtained by combining the ideas in Ref. [GG] with the ones introduced in Ref. [Ge]. In Sect. 5 we discuss the non-resonant resummations while the new infrared resummations are studied in Sect. 6 where a “fully renormalized series” is obtained, i.e. a resummation of the series defining the formal expansion of the quasi-periodic solution (1.6) of the equations of motion (1.5), which we prove to be absolutely convergent. The resummations that we have to perform are really of divergent series. resummations p , with |z| > 1. In particular They concern sums of geometric series of the form ∞ z p=0 among the (infinitely many) cases that we really meet there is the following rule which we use in an essential way: ∞
2+p = −1.
(1.8)
p=0
The paper is a self-contained discussion of the construction and of the convergence of the resummed series. This includes a self-contained description of the well-known formal series [JLZ, GG]. Once this is achieved one has to check that the defined functions do actually represent parametric equations of invariant tori: for this we follow, in Appendix A5, the analysis of Refs. [GG, Ge]. 2. Tree Formalism We look for a formal power series expansion (in ε) of the parametric equations h = (a, b) of the invariant torus close to the torus α = ψ, β = β 0 , h(ψ; ε) =
∞ k=1
ε k h(k) (ψ) =
ν∈Zr
eiν·ψ hν (ε) =
∞ k=1
εk
ν∈Zr
eiν·ψ hν(k) ,
(2.1)
where we have not explicitly written the dependence on β 0 . The power series is easy to derive, see for instance Ref. [GG]: however its convergence turns out to be substantially harder to prove than the convergence of the Lindstedt series for the maximal KAM tori. The series constructed below for our problem, which we still call Lindstedt series, is (k) naturally described in terms of trees. The coefficients hν can be computed as sums of “values” that we attribute to trees whose nodes and lines carry a few labels, which we call “decorated trees”. The formalism to define trees, decorations and values has been described many times and used in the proof of several stability results in Hamiltonian mechanics. Usage of graphical tools based on trees in the context of KAM theory has been advocated recently in the literature as an interpretation of Ref. [E1]; see for instance Refs. [Ga1, GG, BGGM, BaG]. A tree θ (see Fig. 1) is defined as a partially ordered set of points, connected by oriented lines. The lines are consistently oriented toward the root, which is the leftmost point r; the line entering the root is called the root line. If a line connects two points def
v1 , v2 and is oriented from v2 to v1 , we say that v2 ≺ v1 and we shall write v 2 = v1 def
and v2 = ; we shall say also that exits v2 and enters v1 . More generally we write
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323
Fig. 1. A tree θ with 12 nodes; one has pv0 = 2, pv1 = 2, pv2 = 3, pv3 = 2, pv4 = 2. The length of the lines should be the same but it is drawn of arbitrary size. The nodes vi , i = 5, . . . , 11 will be called endnodes. The separated line illustrates the way to think of the label η = (γ , γ )
v2 ≺ v1 if v1 is on the path of lines connecting v2 to the root: hence the orientation of the lines is opposite to the partial ordering relation ≺. The points different from the root will be called the nodes of the tree. Each line from v to v carries a pair η of component labels η = (γ , γ ) ranging in {1, . . . , d} (marked in Fig. 1 only on some of the lines for clarity of the drawing). The labels γ and γ should be regarded as associated with v and v , respectively; hence with each node v with pv entering lines 1 , . . . , pv one can associate pv + 1 labels γ0 , γ1 , . . . , γpv , with γ0 = γv and γj = γ j . Also the root line (from v0 to the root) carries two such labels and the one associated with the final extreme of the root line will be called the root label. Fix any v ∈ θ , we shall say that the subset of θ containing v as well as all nodes w v and all lines connecting them is a subtree of θ with root v ; of course a subtree is a tree. Given a tree, with each node v we associate a harmonic or mode, as called in Ref. [GG], i.e. a label ν v ∈ Zr . We shall denote by V (θ ) the set of nodes and by (θ ) the set of lines. The number k = |V (θ )| of nodes in the tree θ , equal to the number |(θ )| of lines, will be called the order of θ . We call a node with one entering line and 0 harmonic label a trivial node. With any line = v we associate (besides the above mentioned pair η = (γ , γ ) of labels assuming values in {1, . . . , d}) a momentum ν ∈ Zr defined as ν ≡ ν v =
νw.
(2.2)
w∈V (θ ) w v
We shall consider only trees not containing trivial nodes with the entering line with 0 momentum: this is an important restriction, as we shall see, which is a consequence of the derivation of the Lindstedt series, see Ref. [GG] and the comments at the end of this section. We call degree P (θ ) of a tree the order of the tree minus the number of 0 momentum lines, so that |V (θ)| − P (θ ) is the number of the latter.
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We call ν,k,γ the set of trees θ whose root line v0 has momentum ν, root label γ and have order k, i.e. with |V (θ )| = k nodes, while we call oν,k,γ the set of trees of degree k, i.e. with P (θ) = k. One has ν,k,γ = oν,k,γ . Each tree θ “decorated” by labels in the described way will have a value which is defined in terms of a product of several factors. • With each node v we associate a node factor ∂γj fν v (β 0 ), (2.3) Fv = j
where the labels γj are the pv + 1 labels associated with the extreme v of the pv lines entering the node v and of the line exiting it, and the derivatives ∂γ , with γ = 1, 2, . . . , r, have to be interpreted as factors (iν v )γ . Hence Fv is a tensor of rank pv + 1. • With each line carrying labels η = (γ , γ ) and momentum ν we associate a matrix, called propagator, G ≡ δγ ,γ
1 , (ω · ν )2
if ν = 0,
G ≡ −ε −1 (∂β2 f0 (β 0 ))−1 χ (γ , γ > r), γ ,γ
if ν = 0,
(2.4)
where χ (γ , γ > r) is 1 if both γ and γ are strictly greater than r, and 0 otherwise. Given the definitions (2.3) and (2.4) define a value function Val, which with each tree θ of order k associates a tree value εk Fv G , (2.5) Val(θ ) = k! v∈V (θ)
∈(θ)
where, by the definitions, all labels γi associated with the nodes appear twice because they appear also in the propagators: we make in (2.5) the summation convention that repeated γ labels associated with nodes and lines are summed over, with the exception of the label γ associated with the root (because we do not consider it a node and the corresponding label γ appears only once in (2.5). Therefore (2.5) is a number labeled by γ = 1, . . . , d, i.e. Val(θ ) is a vector. Remarks. (1) The trees can be drawn in various ways: we can limit the arbitrariness by demanding that the length of the segments representing the lines is 1 (unlike the drawings in the above figures) and that the angles between the lines are irrelevant. The combinatorics being very important, because it matters in the check of cancellations essential for the analysis, we adopt the convention that trees are drawn on a plane, and that their lines have unit length and carry an identifier label, that we call a number label (not shown in the above figures) which distinguishes the lines from each other even if we ignore the other labels attached to them. Furthermore two trees that can be superposed by pivoting the lines merging in the same node v, around v itself, are considered identical. This is a convention which is useful for checking cancellations; however it is by no means the only possible one. Others are possible and often very convenient in other respects [Ga1, GM1], but in a given work a choice has to be made once and for all. Depriving trees of harmonic and component labels leaves us therefore k k−1 numbered trees (Cayley’s formula). (2) A line carrying 0 momentum is somewhat special. We could visualize the part of the tree preceding such lines by encircling it into a dotted circle: such a representation
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325
has been used in earlier papers, e.g. in Ref. [GG], calling the subtree θ with as root line a leaf. Here, however, we shall avoid using a special word for the 0 momentum lines and the subtrees preceding them. (3) We can think of the propagators as matrices of the form G,αα G,αβ , (2.6) G = G,βα G,ββ where G,αα , G,αβ , G,βα and G,ββ are r × r, r × s, s × r and s × s matrices. (4) The value of a tree θ defined above has no pole at ε = 0 if Val(θ ) = 0 because every line with 0 momentum is preceded by at least two nodes, so that the total power of ε to which the value is proportional is always non-negative and, in fact, it is necessarily positive; we need to take into account that ∂β f0 (β 0 ) ≡ 0 and that our trees contain no trivial nodes with one entering line with 0 momentum. Note that Val(θ ) is a monomial in ε of degree P (θ). (5) In the case of maximal tori and if Val(θ ) = 0 there are no lines with 0 momentum for systems described by the Hamiltonians (1.1): indeed s = 0, see also Ref. [Ga2]. In this case the number of nodes, i.e. the tree order, coincides with the power of ε associated with the monomial in ε defined by the tree value, i.e. with the tree degree. In general, however, the order |V (θ)| of a tree can be larger than its degree P (θ ): |V (θ )| ≥ P (θ) ≥ 21 |V (θ )|. The above definitions uniquely attribute a value to each tree. The following result states the existence of formal solutions to (1.5) which are conjugated to the unperturbed motion, i.e. of the form (1.6), with ψ → ψ + ω0 t, provided the value β 0 is suitably fixed. The proof is an algebraic check which does not distinguish the possible signs of ε and can be taken from Ref. [GG] where it is done in the case ε < 0. Lemma 1. The Fourier transform of the power series solution h = (a, b) of (1.5) of the form (2.1) is obtained by writing (the definition of ok,ν,γ follows (2.2)) εk h(k) Val(θ ) (2.7) ν,γ = θ∈ ok,ν,γ
for all ν ∈ Zr , all k ∈ N and γ = 1, . . . , d. The expression (2.7) is well defined at fixed k and the sum over k gives what we call the formal power series solution for the equations for the parametric representation (2.1), (1.6) of the invariant tori. Note that the formal solubility of the Eqs. (1.5) requires that to each order k one has (k) (k) (k) ∂β2 f0 (β 0 )b0 +R0 = 0, where R0 denotes all the other contributions to order k (which (k )
depend on b0 only with k < k). The first term would be represented by a tree with a triv(k) (k) ial node with entering line carrying zero momentum. If we set b0 = (∂β2 f0 (β 0 ))−1 R0 then we exclude the possibility of such trivial nodes (see comments after (2.2) and auto(k) matically define b0 according to (2.5) with the definition in the second line of (2.4) for propagators of lines with 0 momentum. 3. The Simplest Resummation
The power series in ε in (2.1) and its Fourier transform defined by the sum over k of (2.7) may be not convergent as a power series (as far as we know). The problem is difficult because if in (2.7) we replace Val(θ ) with |Val(θ )| the series certainly diverges.
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Our aim, as stated in the introduction, is to show that nevertheless a meaning to the series can be given. We shall show that the tree values can be further decomposed into sums of several other quantities and that the various contributions to the series can be rearranged by suitably collecting them into families: the sums of the contributions from each family leave us with a new series (no longer a power series in ε) which is in fact convergent and its sum solves the problem of constructing the parametric representations h = (a, b), (2.1), of the invariant tori at least for all ε ∈ E, with E a set with 0 as a density point (i.e. a Lebesgue point). For this purpose we need to define and consider more involved trees and more involved definitions of their values. We begin by remarking that trees may contain trivial nodes, i.e. nodes with 0 harmonic separating two lines with equal momentum ν = 0. One can suppose that no tree contains trivial nodes provided we use for all lines, with momentum ν = 0 and labels γ , γ associated with the extremes, the new propagators def
g(x; ε) = (x 2 − M0 )−1 ,
def
def
x = ω · ν,
M0 = ε
0 0 . 0 ∂β2 f0 (β 0 )
(3.1)
This is a resummation of many divergent series obtained by adding the values of trees obtained from a tree without trivial nodes by “insertion” of an arbitrary number of trivial nodes on the branches with momentum ν = 0: this requires summing series, one per branch of a tree without trivial nodes, which are geometric series with ratio given by M0 the d × d matrix z = (ω·ν) 2 ; |z| can be larger than 1 because the s non-zero eigenvalues εaj , j = 1, . . . , s, of M0 are unrelated to x = ω · ν.1 p −1 is not rigorous and needs to be eventually Therefore replacing ∞ p=0 z by (1 − z) justified. Certainly we must at least suppose that x 2 − M0 can be inverted: otherwise the values of the trees representing the new series might even be meaningless! (i.e. if some lines will have momentum ν such that det(x 2 − M0 ) = 0). This happens for a dense set of ε’s and we have to exclude such ε’s by imposing conditions on the eigenvalues λ[0] r+j ≡ εaj , j = 1, . . . , s, i.e. on ε. For uniformity of notations it is convenient to assume that ε is in an interval (εmin , 4εmin ] related to the largest eigenvalue λ[0] d ≡ as ε of M0 by def
λ[0] d ≡ ε as ∈ IC =
1 4
C2, C2 ,
def
C = C0 2−n0 ,
n0 ≥ 0,
(3.2)
where C0 is the Diophantine constant in (1.3) (fixed throughout the analysis); thus IC is an interval of size O(C 2 ) (i.e. 43 C 2 ). In other words we find it convenient to measure ε in units of C02 as−1 via an integer n0 . We a priori assume, for simplicity, the restrictions as ε ≤ C02 and ε ≤ 1. To give a meaning to (x 2 − M0 )−1 it would suffice to require |x 2 − εaj | = 0 for all j and all ν, thereby excluding “only” a denumerable (dense) set of values of ε, of 0 length; however stronger conditions will be needed in order to analyze the convergence problems and we begin by imposing them in a form which will be useful later. Setting 1 Note that since the tree lines are numbered (i.e. they are regarded as distinct) adding p nodes on a line changes the combinatorial factor k!−1 in (2.5) into (k + p)!−1 ; however the new p lines thus
produced can be chosen in k+p ways and ordered in p! ways so that we can ignore the extra number p
−1 labels on and use as combinatorial factor (k + p)!−1 k+p p p! = k! .
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327
def
[0] [0] for later use λ[0] j (ε) = λj , the conditions that we impose on λj , i.e. on ε, are that for all x = ω · ν = 0 and for all independent choices of the signs + or −,
C0 def [0] [0] (x) = min |x| − λ[0] (ε) , ± λ (ε) ± λ (ε) x ≥ 2−(n0 −1)/2 τ1 , j j i j ≥i |ν| (3.3)
for τ1 suitably large and n0 suitably larger than n0 , see (4.2). This excludes a closed set of values of ε in the considered interval IC , (3.2): its measure can be estimated without difficulties. Let τ1 = τ0 + r + 1,
(3.4)
be a convenient, although somewhat arbitrary, choice; then the total measure of the excluded set is ≤ 2−(n0 −1)/2 C 2 K,
(3.5)
where K is a suitable constant; see Appendix A2. Hence the measure of the complement of the set En0 −1 where (3.3) is verified is a small fraction of order C 1/2 of the measure of the interval IC , whose size is proportional to C 2 , in which we let ε vary, at least if n0 is large. 4. Resummations: Semantic and Heuristic Considerations Replacing the propagators x −2 of the lines by (x 2 − M0 )−1 we obtain a representation of the parametric equations h involving simpler trees (i.e. trees with no trivial nodes). The new representation is a series in which each term is well defined if ε is in the large set En0 −1 ⊂ IC in which (3.3) holds. This is quite different from the original Lindstedt series in (2.7) whose terms are well defined for all ε. We should also stress that the resummed series is in a sense more natural: the 0 momentum lines now appear as less anomalous because their propagator is much more closely related to (x 2 − M0 )−1 . One can say that it is just the latter evaluated at x = 0 with the meaningless entries (i.e. the first r diagonal entries) replaced by 0. Another way of saying the latter property is that lines with 0 momentum and labels γ , γ ≤ r are forbidden. One should not be surprised by this fact: it is the generalization of the corresponding property in the case of maximal tori (r = d) in which this means that lines with 0 momentum are forbidden. The latter property goes back to Poincar´e’s theory of the Lindstedt series and is the key to the proof of the KAM theorem and of cancellations which make the formal Lindstedt series for maximal tori absolutely convergent; see Refs. [E1, Ga2]. However the new series is still only a formal representation because it is by no means clear that it is absolutely convergent. The next natural idea is to try to establish convergence by further modifying the propagators, changing at the same time the trees structure, until one achieves a formal representation whose convergence will be “easy” to check. Once we have achieved a formal representation which is convergent we shall have to check that it really solves the equations for h. The modification of the trees structure will be performed by steps. At each step, labeled by an integer n = 0, 1, . . . , the propagators of the lines with non-zero momentum will have been modified acquiring labels [0], [1], . . . [n − 1], or the label [≥ n],
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indicating that they are given no longer by (x 2 − M0 )−1 but by a matrix proportional to (x 2 − M[≤p] )−1 , if their label is [p], with p < n, or (with a different proportionality factor) to (x 2 − M[≤n] )−1 , if their label is [≥ n]; here M[≤p] are suitable matrices. Here and in the following the symbols [≤ n] and [≥ n] are consistently used. Hence [≥ n] does not denote the set of scales [p] with p ≥ n, and in fact it is just a different scale; likewise [≤ n] does not “include” [p] even if p ≤ n. In other words one has to regard the symbols [≤ n], [n] and [≥ n] as unrelated symbols. This might appear unusual but it turns out to be a good notation for our purposes. The proportionality factor depends on x and contains cut-off functions which vanish unless x 2 − M[≤p] has smallest eigenvalue of order O(2−2p C02 ); the cut-offs are so devised that if the propagator does not vanish its denominator has a minimum size proportional to 2−2p and the ratio between its minimum and maximum values will be bounded above and below by a p-independent constant. No modification will be made of the propagators of the 0 momentum lines; for uniformity of notation we shall attach a label [−1] to such lines. Considering trees with no trivial nodes in which each line carries also an extra scale label [−1], [0], [1], . . . [n − 1], [≥ n] a new formal representation of h will be obtained by assigning, to the trees, values defined by the same formula in (2.5), with the propaga[p] tors G replaced by the new propagators, that we denote g if the line carries the label [p], with p = −1, 0, . . . , n − 1, and g[≥n] if the line carries the label [≥ n]. When the line is on scale [p], with p = 0, . . . , n − 1, or [≥ n] or [−1], then the corresponding 2 [≤p] )−1 or (x 2 − M[≤n] )−1 or, see (2.4), the propagator will be proportional to (x −M 0 0 block matrix . 0 (−ε∂β2 f0 (β 0 ))−1 The construction will be performed in such a way that the matrices (x 2 − M[≤p] ) will be defined by series which will be proved to be convergent; furthermore if we only considered the contributions to the formal representation of h coming from trees in which no propagator carries the “last label” [≥ n] then the corresponding series would be convergent. We express the latter property by saying that the performed resummations regularize the formal representation of h down to scale [n − 1], or that the propagators singularities are probed down to scale [n − 1]. The problem of course remains to understand the contributions from the trees containing lines with label [≥ n]. The construction will be such that their propagators are also properly defined because the matrices M[≤n] will always be well defined by convergent series (as we shall see). However for the lines whose label is [≥ n] no useful positive lower bound, not even n-dependent, can be given on the smallest eigenvalue of the denominators in the corresponding propagators. We shall say that the lines with scale [≥ n] probe the singularity all the way down to the smallest frequencies or all the way down in the infrared scales. Thus in spite of the convergence of the contributions to h coming from trees with labels [−1], [0], [1], . . . , [n − 1] the representation of h remains formal. Therefore we shall proceed by increasing the value of n trying to take the limit n → ∞. This is the procedure followed in the case of the theory of hyperbolic tori in Ref. [GG]. In that case, however, the propagators denominators (x 2 − M[≤n] ) had eigenvalues always bounded below proportionally to x 2 . Indeed the last s eigenvalues of M[≤n] were negative whereas the first r remained close to zero within O(εx 2 ) (a non-trivial property, however, due to remarkable cancellations well known in the KAM theory [Ga2]).
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Here the matrices x 2 − M[≤n] will be shown to have the first r eigenvalues differing by a factor (1 + O(ε 2 )) and the last s differing by O(ε 2 ) with respect to those of x 2 − M0 (which has by construction r eigenvalues x 2 and s eigenvalues x 2 − εaj j = 1, . . . , s). Thus the denominators can become small because either x 2 gets close to 0 or because it gets close to εa1 , . . . εas . Therefore the regularization will have to be split in two parts. The first part will concern regularizing the scales [p] with p such that the eigenvalues of x 2 − M[≤p] remain bounded below proportionally to x 2 ; we shall call this part of the analysis the high frequencies resummation. The other part, which we shall call the infrared resummation, will concern the regularization of the scales [p], in which x can be so close to some εaj that the denominators cannot be bounded below proportionally to x 2 . We associate with each momentum ν the frequency x = ω · ν and we measure the strength of this resonance by the integer p if D(x; ε) C02 2−2p , with def 2 [0] D(x; ε) = min x 2 − λ[0] (ε) − λ (ε) (4.1) = . x j jε (x) j
Therefore the condition that the resonance strength of the frequency x be bounded below proportionally to |x| is that p is not too large compared to n0 defined in (3.2), so that x 2 stays away from the corresponding eigenvalue λ[0] j (ε) by more than a small fraction of the minimum separation δ between the distinct eigenvalues. For instance we can require D(x; ε) ≥ 2−2(n0 +1) C02 ≥ δ/4. This gives p ≤ n0 , with n0 = n0 + n,
def
n = −1+
1 1 log2 , 2 ρ
ρ=
1 −1 a min{a1 , min{aj +1 − aj }}. j 4 s (4.2)
In fact the requirement could be fulfilled with n one unit larger: the interest of using the above value of n will emerge later (if s = 1 one interprets ρ = 41 ). We then perform the analysis by defining recursively the matrices M[≤p] (x; ε) for [p] p = 0, . . . , n0 with eigenvalues λj (x, ε) verifying for a suitable constant γ > 0, [p]
2 |λj (x, ε) − λ[0] j (ε)| < γ ε ,
p ≤ n0 ,
(4.3)
so that if the label p of the line with frequency x is p ≤ n0 then one has, if 21 as 2−2(n+1) − γ ε ≥ 0, [p]
|x 2 − λj (x, ε)| ≥
1 1 1 D(x, ε) + D(x, ε) − γ ε 2 ≥ D(x, ε) ≥ 2−2(n+2) |x|2 , 2 2 2 (4.4)
where the last step is obvious if |x|2 ≥ 2λ[0] d (ε), otherwise it follows from the inequality −2(n+1)−1 |x|2 . D(x; ε) ≥ 2−2(n+1) 2−2n0 C02 ≥ 2−2(n+1) λ[0] d (ε) ≥ 2
(4.5)
We can say that for p ≤ n0 the strength of the singularity is dominated by the distance |x| to the origin, i.e. by the “classical” small divisors x −2 provided, of course, the matrices x 2 − M[≤p] (x; ε) remain close enough to x 2 − M0 (which we shall check). Furthermore the convergence of the sum of all values of trees with no line label [≥ n0 ] will be performed exactly along the lines of Ref. [GG] because the bound (4.4) guarantees that
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in evaluating such trees one does not probe the singularities close to the eigenvalues of M0 . The departure from the method in Ref. [GG] occurs when we consider trees in which lines bear the label [≥ n0 ]. The problem will again be studied by a multiscale analysis which will have to be suitably modified to allow probing
the new singularities arising λ[0] j (ε), j > r. The difficulty
is that the propagator g[≥n0 ] will not be singular exactly at the frequencies λ[0] j (ε) = 0
from the resonances between the frequencies x and the
0] but at the frequencies fixed by the roots of the eigenvalues λ[≤n (x; ε) of the matrices j [≤n ] 0 M (x; ε). The latter not only are slightly different from those of M0 but will turn out to depend also on x. 0] (x; ε) no longer This means that D(x; ε) and even (x; ε) = minj x 2 − λ[≤n j provide a good estimate of the strength of the singularity, because D, vanish at the “wrong places”. In fact we shall have to perform a multiscale analysis to resolve the infrared singularities, and it will happen that at each of the new scales with labels [p], with p ≥ n0 , the singularities will keep moving. Suppose, we have regularized the series up to scale [n − 1], with n > n0 , introducing suitably matrices M[≤p] (x; ε), with p = n0 , . . . , n, thus pushing the probe of the singularities down to scales C0 2−n ; then to avoid meaningless expressions we shall have to impose on the eigenvalues of the last propagator, proportional to (x 2 − M[≤n] (x; ε))−1 , a condition like (3.3). Since the eigenvalues depend on n and x this risks to imply that we have to discard too many ε’s; in the limit n → ∞: when, finally, the singularities will have been probed on all scales, or even for large enough scales, we might be left with an empty set of ε’s rather than with a set of almost full measure. Physically the difficulty shows up because of the possibility of resonances between the proper frequencies of the quasi-periodic motion on the tori and the normal frequencies. It will be studied and solved in Sect. 6 below, while in Sect. 5 we shall discuss the simpler regularization of the series for h on the high frequency scales. The spirit informing the analysis is very close to the techniques used in harmonic analysis, in quantum field theory and in statistical mechanics, known as “renormalization group methods” (see Refs. [F, Ga3, GM1, Ga4, BKS, Ga5]). The latter methods are also based on a “multiscale decomposition” of the propagator’s singularities. We introduced and adopted the above terminology because we feel that it is suggestive and provides useful intuition at least to the readers who have some acquaintance with the renormalization group approach and multiscale analysis.
5. Non-Resonant Resummations The resummations will be defined via trees with no trivial nodes and with lines bearing further labels. Moreover the definition of propagator will be changed, hence the values of the trees will be different from the ones in Sect. 3: they are constructed recursively. Instead of the sharp multiscale decomposition considered in Ref. [GG] here it will be convenient to work with a smooth one as in Ref. [Ge]. Let ψ(D) be a C ∞ non-decreasing compact support function defined for D ≥ 0, see Fig. 2, such that ψ(D) = 1,
for
D ≥ C02 ,
ψ(D) = 0,
for
D ≤ C02 /4,
(5.1)
where C0 is the Diophantine constant in (1.3), and let χ (D) = 1 − ψ(D). Define also ψn (D) = ψ(22n D) and χn (D) = χ (22n D) for all n ≥ 0. Hence ψ0 = ψ, χ0 = χ and
Degenerate Elliptic Resonances
331
Fig. 2. The first graph is ψ0 , the second is χ0 and the third is χ 0 = ψ1 χ0
1 ≡ ψn ((x; ε)) + χn ((x; ε)),
for all n ≥ 0,
(5.2)
for all choices of the function (x; ε) ≥ 0: in particular for (x, ε) = D(x) with D(x) defined in (5.3) below. We set the following notations. Definition 1. Let n0 , n be as in (4.2) and D(x; ε) as in (4.1). (i) Divide the interval IC ≡ [εmin , 4εmin ], where ε varies, see (3.2), into a finite number of small intervals I of size 21 εmin ρ (or smaller), see (4.2), i.e. smaller than a fraction of the minimum separation between the eigenvalues 0, a1 , . . . , as . Define 2 [0] (ε) = min − λ (ε) (5.3) D(x; I ) = min D(x; ε) = min min x 2 − λ[0] x , j j (x) ε∈I
ε∈I
j
ε∈I
where j (x) is the smallest value of j for which the last equality holds: exceptionally there might be 2 such labels. The j (x) is ε-independent, by construction, for ε ∈ I . Remarks. (1) Note that, as a consequence of the definition of the intervals I and of D(x; I ) as given by (5.3), one has, for all ε ∈ I , 1 (ε) ≥ x 2 − λ[0] (ε) . (5.4) min x 2 − λ[0] j j (x) j 2 (2) If ε is in one of the intervals I and x verifies D(x; I ) ≤ C02 2−2n0 then there is only one value of j for which the last equality in (5.3) holds. (3) We shall fix, from now on, ε in one of the intervals I ⊆ IC . Remark that D(x; I ) is piecewise linear in x 2 with slope equal to 1 in absolute value for x in the regions where it will be considered (see below) and we simplify the notation by setting def
D(x) = D(x; I ).
(5.5)
(4) The number of intervals I ⊂ IC can and will be taken independent of εmin , i.e. of the interval IC where ε varies, and equal to a fixed integer of order 6ρ −1 . (5) From now on we only consider trees with no trivial nodes. A simple way to represent the value of a tree as the sum of many terms is to make use def
of the identity in (5.2). The propagator g(x; ε) ≡ g [≥0] (x; ε) = (x 2 − M0 )−1 of each line with non-zero momentum (hence with x = 0) is written as g [≥0] (x; ε) = ψ0 (D(x)) g [≥0] (x; ε) + χ0 (D(x)) g [≥0] (x; ε) def [0] = g (x; ε) + g ≥1 (x; ε), and we note that
g [0] (x; ε)
vanishes if D(x) is smaller than (C0
/2)2 ,
see Fig. 2.
(5.6)
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If we replace each g [≥0] (x; ε) with the sum in (5.6) then the value of each tree of order k is split as a sum of up to 2k terms2 which can be identified by affixing on each line with momentum ν = 0 a label [0] or ≥ 1 . Further splittings of the tree values can be achieved as follows. [p]
Definition 2. For p = 1, . . . , n0 , let M[≤p] (x; ε) be matrices with eigenvalues λj (x; ε), p = 1, . . . , n; we set M[0] (x; ε) ≡ M0 and M[≤n] (x; ε) = np=0 M[p] (x; ε). Define for 0 < n ≤ n0 − 1, ψn (D(x)) n−1 m=0 χm (D(x)) , 2 x − M[≤n] (x; ε) n−1 χm (D(x)) def ≥n g (x; ε) = 2 m=0 [≤n−1] , x −M (x; ε) n−1 χm (D(x)) def g [≥n] (x; ε) = 2 m=0 [≤n] , x −M (x; ε) def
g [n] (x; ε) =
(5.7)
and g [0] (x; ε) = ψ0 (D(x)) (x 2 − M0 )−1 . We call the labels [n], {≥ n}, [≥ n] scale labels. Remarks. (1) The products n−1 m=0 χm (D(x)) can be simplified to involve only the last factor: we keep the notation above as it is a notation that naturally reflects the construc-
tion. The propagators g ≥n play a subsidiary role and are here for later reference. (2) The matrices M[p] (x; ε) will be defined recursively under the requirement that the functions h defining the parametric equations of the invariant torus will be expressed in terms of trees whose lines carry scale labels indicating that their values are computed with the propagators in (5.7). (3) Note that if we defined M[≤p] (x; ε) ≡ M0 , i.e. M[p] ≡ 0 for p > 0, then (recall that we consider only trees without trivial nodes) we would naturally decompose (see below for details) the tree values into sums of many terms keeping obviously each total sum constant by repeatedly using (5.2), thus meeting the requirement in Remark (2) above. This would be of no interest. Therefore we shall try to define the matrices M[p] (x; ε) so that the sum of the values of new trees (with no trivial nodes and whose nodes and lines still carry harmonic and momentum labels as well as scale labels [−1], [0], . . . , [n − 1], [≥ n]) remain the same provided their values are evaluated by using the propagators in (5.7) and we shall try to define M[≤p] (x; ε), so that there is also control of the convergence. (4) In other words we try to obtain a graphical representation of h, involving values of trees which are easier to study at the price of needing more involved propagators. This is a typical method employed in KAM theory [GBG], and in other fields. To define recursively the matrices we introduce the notions of clusters and of selfenergy clusters of a tree whose lines and nodes carry the same labels introduced so far and in addition each line carries a scale label which can be either [−1], if the momentum of the line is zero, or [p], with p = 0, . . . , n0 − 1, or [≥ n0 ]. Given a tree θ decorated in this way we give the following definition, for n ≤ n0 . 2
Not necessarily 2k because there might be lines on scale [−1] whose propagator is not decomposed.
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333
Definition 3 (Clusters). (i) A cluster T on scale [n], with 0 ≤ n, is a maximal set of nodes and lines connecting them with propagators of scales [p], p ≤ n, one of which, at least, of scale exactly [n]. We denote with V (T ) and (T ) the set of nodes and the set of lines, respectively, contained in T . The number of nodes in T will define the order of T , and it will be denoted with kT . (ii) The mT ≥ 0 lines entering the cluster T and the possible line coming out of it (unique if existing at all) are called the external lines of the cluster T . (iii) Given a cluster T on scale [n], we shall call nT = n its scale. Remarks. (1) For instance if n = 0 the scale of the lines in the cluster can only be [−1], [0]. Note that a single node is not a cluster. Also connected subgraphs containing only lines on scales [−1] are not clusters, because by definition the scale [n] of the cluster has to such that n ≥ 0. (2) Here n ≤ n0 − 1. However the definition above is given in such a way that it will extend unchanged when also scales larger than n0 are introduced. (3) The clusters of a tree can be regarded as sets of lines hierarchically ordered by inclusion and have hierarchically ordered scales. (4) A cluster T is not a tree (in our sense). However we can uniquely associate a tree with it by adding the entering and the exiting lines and by imagining that the lower extreme of the exiting line is the root and that the highest extremes of the entering lines are nodes carrying a harmonic label equal to the momentum flowing into them; see Fig. 3. Definition 4 (Self-energy clusters). (i) We call self-energy cluster of a tree θ any cluster T of scale [n] such that T has only one entering line 2T and one exiting line 1T , and furthermore v∈V (T ) ν v = 0. (ii) The order of a self-energy cluster is the number of nodes. Remark. The essential property of a self-energy cluster is that it has necessarily just one entering line and one exiting line, and both have equal momentum (because v∈V (T ) ν v = 0). Note that both scales of the external lines of a self-energy cluster T are strictly larger than the scale of T as a cluster, but they can be different from each other by at most one unit. Furthermore the degree of a self-energy cluster is ≥ 2. Of course no self-energy cluster can be on scale [−1] (by definition).
Fig. 3. Illustration of a cluster and of the content of Remark (4): the continuous lines are lines of scale lower than the dashed lines on the right which are lines preceded by an arbitrary subtree; the dashed line on the left ends with a node into which ends an arbitrary subtree and which is continued by another arbitrary subtree (none of the mentioned subtrees is drawn): hence the continuous lines form a cluster (whose lines are surrounded by an ellipse). The cluster itself is depicted in the intermediate figure (manifestly not a tree). The third drawing shows the tree that can be associated with the cluster: the formerly dashed lines are reintroduced and bolder to indicate that they come out of endpoints which have a harmonic label equal to the total momentum flowing in the formerly dashed lines
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Definition 5 (Self-energy matrices). (i) Let R k,ν,γ be the set of trees of degree k with root line momentum ν and root label γ which contain neither self-energy clusters nor trivial nodes. Such trees will be called renormalized trees. R the set of self-energy clusters of degree k and scale [n] which (ii) We denote with Sk,n do not contain any other self-energy cluster nor any trivial node; we call them renormalized self-energy clusters on scale n. R we shall define the self-energy value of T as (iii) Given a self-energy cluster T ∈ Sk,n the matrix3 VT (ω · ν; ε) =
ε k [n ] g Fv , (k − 1)!
(5.8)
v∈V (T )
∈(T )
where g[n ] = g [n ] (ω · ν ; ε). Note that, necessarily, n ≤ n. The kT − 1 lines of the self-energy cluster T will be imagined as distinct and to carry a number label ranging in {1, . . . , kT − 1}. The recursive definition of the matrices M[n] (x; ε), n ≥ 1, will be (if the series converges) M (x; ε) = [n]
n−1
∞ χp (D(x))
p=0
def
VT (x; ε) =
n−1
k=2 T ∈S R
χp (D(x)) M [n] (x; ε),
p=0
k,n−1
(5.9) where the self-energy values are evaluated by means of the propagators on scales [p], with p = 0, . . . , n, which makes sense because we have already defined the propagators on scale [0] and the matrices M[0] (x; ε) ≡ M0 (cf. Definition 2). With the above new definitions we have the formal identities hν,γ =
∞
Val(θ ),
(5.10)
k=1 θ∈ R
k,ν,γ
where we have redefined the value of a tree θ ∈ R k,ν,γ as Val(θ ) =
εk [η ] g (ω · ν ; ε) Fv , k! ∈(θ)
(5.11)
v∈V (θ)
with [η ] = [−1], [0], . . . , [n0 − 1], [≥ n0 ]. Note that (5.10) is not a power series in ε. The statement in (5.10) requires some thought, but it turns out to be a tautology, see also Ref. [GG], and Ch. VIII in Ref. [GBG], if one neglects convergence problems which, however, will occupy us in the rest of this paper. A sketch of the argument is as follows. Imagine that we have only scales [−1], [0], . . . , [n − 1], [≥ n], i.e. we have performed the scale decomposition of the propagators up to scale [n − 1] and we have not 3 This is a matrix because the self-energy cluster inherits the labels γ , γ attached to the endnode of the entering line and to the initial node of the exiting line.
Degenerate Elliptic Resonances
335
decomposed the propagators on scale [≥ n] and that we have checked the statement (5.9) and (5.10) (trivially true for n = 0). 0, . . . , n−1 or [≥ n] Given a tree θ ∈ R k,ν,γ with lines carrying labels [p] with p = [n] (x; ε)+g ≥n+1 (x; ε) as in (5.6) as g or [−1], we can split the propagators g [≥n] (x; ε)
with g [n] (x; ε) = ψn (D(x))g [≥n] (x; ε) and g ≥n+1 (x; ε) = χn (D(x))g [≥n] (x; ε). In this way we get new trees which in general contain self-energy clusters of scale [n]. We can in fact construct infinitely many trees with self-energy clusters of scale [n] simply by inserting an arbitrary number of them on any line with scale {≥ n + 1}. The values of the trees obtained by q ≥ 0 such self-energy insertions on a given line of in fact they differ only by a a tree in R k,ν,γ can be arranged into a geometric progression: factor equal to the value of the integer power q in g ≥n+1 (x; ε) M [n+1] (x; ε)g ≥n+1
q+1 if M [n+1] (x; ε) is defined as in (5.9), where the VT (x; ε) are evaluated by (x; ε) using as propagators g [p] (x; ε), with 0 ≤ p ≤ n or p= −1, for the lines carrying a
scale label [p]. Summation over q will simply change g ≥n+1 (x; ε) into g [≥n+1] (x; ε) and at the same time one shall have to consider only trees with no self-energy cluster of scale [n] nor of scale [p] with p < n and with lines carrying scale labels [−1], . . . , [n] or [≥ n+1]. In this way we prove (5.10) for all n ≤ n0 −1 (in particular for n = n0 −1). We could continue, but for the reasons outlined in Sect. 4, we decide to stop the resummations at this scale. In other words the above is a generalization of the simple resummation considered in Sect. 3. The result is still as formal as the Lindstedt series we started with, even assuming convergence of the series in (5.9). In fact the consequent expression for h cannot even be, if taken literally, correct because as in Sect. 3 the denominators in the new expressions could even vanish because no lower cut-off operates on the lines with scale [≥ n0 ] as the third of (5.7) shows. To proceed we must first check that the series (5.9) defining M [n] (x; ε) are really convergent. In spite of the last comment this will be true because in the evaluation of M [n] (x; ε) the only propagators needed have scales [p] with p ≤ n − 1 so that, see the factors ψn (D(x)), χn (D(x)) in (5.7), their denominators not only do not vanish but have controlled sizes that can be bounded below proportionally to x 2 by (4.4), i.e. simply by a constant times C02 |ν|−2τ0 , see (1.3), or by (ε a1 )−1 for the lines with 0 momentum. In Ref. [GG] it has been shown by a purely algebraic symmetry argument that, as long as one can prove convergence of the series in (5.9), the matrices M [n] (x; ε) are Hermitian and (M [n] (x; ε))T = M [n] (−x; ε). Furthermore we should expect that the eigenvalues of the matrix M[≤n] (x; ε) should be approximately located either near 0 or near εa1 , . . . , εas at least within O(ε 2 ); see Fig. 4. The expectation relies on Ref. [GG] (see Eq. (3.25)) where the following “cancellations result” is derived for n0 large enough (hence for ε small because 2−2n0 −2 < εas ≤ 2−2n0 C02 ). We reproduce the proof in Appendix A3 below, adapting it to the present notations.
Fig. 4. The eigenvalues of M[≤n] (0; ε) for n ≤ n0 : the first r of them are below O(ε 2 ), while the remaining s are located near the eigenvalues of the positive definite matrix ∂β2 f0 (β 0 ): εa1 , . . . , εas respectively
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G. Gentile, G. Gallavotti
Lemma 2. If n0 is large enough and n ≤ n0 = n0 + n (see (4.2)) then the following properties hold: (i) The matrices M[≤n] (x; ε), x = ω · ν, are Hermitian and can be written as [≤n] Mαα (x; ε) M[≤n] [≤n] αβ (x; ε) , (5.12) (x; ε) = M [≤n] M[≤n] βα (x; ε) Mββ (x; ε) where the labels α run over {1, . . . , r} and β over {r + 1, . . . , }. (ii) One has M[≤n] (x; ε) = (M[≤n] (−x; ε))T , so that the eigenvalues of M[≤n] (x; ε) [n] 2 4 verify the symmetry property λ[n] j (x; ε) = λj (−x, ε), i.e. they are functions of x . (iii) Let ∂x± be right and left x-derivatives, then M[n] (x, ε) ≤ B ε 2 e−κ1 2 M[n] αα (x; ε) ≤ B M[n] αβ (x; ε) ≤ B M[n] ββ (x; ε)
n/τ
,
−1/2
∂x± M[≤n] (x, ε) ≤ Bε2 as
,
∂ε± M[≤n] (x, ε) ≤ B ε, n/τ e−κ1 2 min{ε 2 , ε x 2 as−1 }, 3 n/τ −1/2 e−κ1 2 min{ε 2 , ε 2 |x| as },
≤ B e−κ1
2n/τ
(5.13)
ε2 ,
for n ≤ n0 and for suitable n0 -independent constants B, κ1 , τ > 0; one can take τ = τ0 . While κ1 is dimensionless the constants A , A, B have the same dimension (of a frequency square): this is the purpose of introducing appropriate powers of as . General properties of matrices and (5.13) imply, see Appendix A4, A < |∂ε λ[n] j (x; ε)| < A,
1
2 as2 |∂x± λ[n] j (x; ε)| < A ε ,
[n] A < |∂ε (λ[n] j (x; ε) − λi (x; ε))|, [n−1] (x; ε)| ≤ ε2 B e−κ1 |λ[n] j (x; ε) − λj
j > r, i = j > r,
2n/τ
,
2 2 −1 |λ[n] j (x; ε)| < A min{ε , ε x as },
(5.14)
j > r, j ≤ r,
where A , A > 0 are n, n0 -independent constants, and τ = τ0 . Remarks. (1) The first three bounds on the eigenvalues in (5.14), follow from the first line of (5.13) by using the self-adjointness of the matrices M[≤n] (x; ε); see Appendix A4. The other bounds in (5.13) imply the last bound in (5.14); see Appendix A4. (2) The natural domain of definition in x of M[n] (x, ε), n > 0, will turn out to be D(x) ≤ 2−2(n−1) C02 , but we imagine that it is defined for all x by continuing it as a constant from its limit value. In fact this is not important because, as we shall see, only the values of M[n] (x, ε) with D(x) ≤ 2−2(n−1) C02 enter into the analysis. Smoothness means differentiability in ε ∈ IC and a right and left differentiability in x. The lack of differentiability in x, but the existence of right and left x derivatives, is due to the For instance if r = s = 2 and f (α , β ) = f0 (β )+f1 (β ) cos α1 +f2 (β ) cos α2 , to lowest order in x, ε, [≤n] [≤n] one has Mαα (x; ε) = 3ε2 x 2 (2ωu4 )−1 [fu2 (β ) + |∂β fu (β )|2 ]δu,v , Mαβ = iε2 x(2ωv3 )−1 ∂βv [(fu2 (β ) + 4
[≤n] |∂β ϕu (β )|2 )], and Mββ = ε∂β2 f0 (β), u, v = 1, 2.
Degenerate Elliptic Resonances
337
fact that the function D(x) admits right and left derivatives: hence lack of differentiability in x appears as an artifact of the method. This lack of smoothness (unpleasant but inessential for our purposes) can be eliminated by changing D(x) into a new D(x) 2 which is smooth for x between successive λj (ε)’s and, at the same time, it is bounded above and below proportionally to D(x). But this would make the discussion needlessly notationally involved and we avoid it. (3) One should also remark that, although we excluded some values of ε (i.e. we required ε ∈ En0 −1 , see (3.3)), here all ε ∈ IC are allowed. The restriction on ε plays no role in the high frequency resummations: so far its only purpose is to avoid divisions by 0 and to assign a finite value to contributions to h from trees with propagators on scale [≥ n0 ] (which could be infinite because of the lack of an infrared cut-off in their expressions; see the third line of (5.7). (4) The bounds on the entries of M[n] (x; ε) in the second and third lines of (5.13) arise from cancellations that are checked in Ref. [GG] via a sequence of algebraic identities on the Lindstedt series coefficients and the real difficulty lies in the proof of convergence. The algebraic mechanism for the cancellations is briefly recalled in Appendix A3, for completeness. (5) Loosely speaking (as mentioned in Sect. 4 the reason why the above result holds with n0 -independent constants, and why its proof can be taken from Ref. [GG], is that if the scales of the propagators are constrained to be [p] with p < n0 the propagators denominators can be estimated by 2−2(n+1)−2 x 2 by (4.4) and by the Remark (1) after Definition 1, or by ε−1 a1 as in [GG] for the lines with 0 momentum. This means that one can proceed as in the hyperbolic tori cases in which boundedness, from below, proportionally to x 2 of the propagators denominators was the main feature exploited and no restriction on ε had to be required, other than suitable smallness. The lemma can be proved by imitating the convergence proof of the KAM theorem, see for instance Ref. [GG]; however in the following Appendix A3 the part of the proof which is not reducible to a purely algebraic check is repeated, for completeness. We have therefore constructed a new representation of the formal series for the function h of the parametric equations for the invariant torus: in it only trees with lines carrying a scale label [−1], [0], . . . , [n0 − 1] or [≥ n0 ] and no self-energy clusters are present. The above lemma will be the starting block of the construction that follows. 6. Renormalization: The Infrared Resummation Convergence problems still arise from the propagators g [≥n0 ] (x; ε), which become uncontrollably large for x = ω · ν close to the eigenvalues of M0 because the bound (4.4) which allowed control of the divisors in terms of the classical small divisors (i.e. in terms of |x|) does not hold any more. Hence we must change strategy. Definition 6. Given d × d Hermitian matrices M[≤n] (x; ε), n = n0 , n0 + 1, . . . , with eigenvalues λ[n] j (x; ε), we introduce the following notations: (i) The sequence of self-energies λ[n] j (ε) is defined for n ≥ n0 by def λ[n] j (ε) =
λ[n] j
λ[n−1] (ε), ε , j
provided λ[n] j (ε) ≥ 0, n = n0 , n0 + 1, . . . .
0 −1] λ[n (ε) = λ[0] j j ,
def
(6.1)
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(ii) The propagator divisors are defined for n ≥ n0 by def [n] (x; ε) = x 2 − λ[n] (ε) , j (x)
(6.2)
where j (x) is the label where the minimum of x 2 − λ[n] (ε) is reached. j Remarks. (1) The self-energies are defined recursively starting from those of the matrix M0 whose first r eigenvalues are 0. Hence, as long as one can extend the last of (5.14) [0] and as long as the self-energies λ[n] j (ε) remain close to the original value λj , as we [n] shall check for ε small enough, one has λ[n] j (ε) = 0 for j = 1, . . . , r and λj (ε) > 0 for j > r. (2) Under the same conditions and if n] (x; ε) 2−2n C02 the label j (x) depends only on M0 , hence it is n-independent, and furthermore it is constant at x fixed, as ε varies in the intervals I introduced in Definition 1 (because for large n the frequency x is constrained to be close to one of the λ[n] j (ε)). (3) The name of propagator divisor assigned to [n] (x, ε) in (6.2) reflects its later use as a lower bound on the denominator of a propagator, see Remark (7) to the inductive assumption below.
By repeating the analysis of Sect. 4 we can represent the function h via sums of values of trees in which lines can carry scale labels [−1], [0], . . . , [n0 − 1], [n0 ], [n0 + 1], . . . and which contain no self-energy clusters and no trivial nodes (i.e. are renormalized trees, see Definition 5 in Sect. 5. The new propagators will be defined by the same procedure used to eliminate the self-energy clusters of scales [n] with n ≤ n0 − 1. However we shall now determine the scale of a line in terms of the recursively defined [n] (x; ε) rather than in terms of D(x): the latter becomes too rough to resolve the separation between the eigenvalues and their variations. def n0 −1 def n [m] (x; ε)) for n ≥ n Let Xn0 −1 (x) = 0 m=n0 χm ( m=0 χm (D(x)), Yn (x; ε) = and Yn0 −1 ≡ 1: the definition of the new propagators will be def
Xn0 −1 (x) ψn0 ([n0 ] (x; ε)) (x 2 − M[≤n0 ] (x; ε))−1 ,
def
Xn0 −1 (x) χn0 ([n0 ] (x; ε)) ψn0 +1
g [n0 ] = g [n0 +1] =
([n0 +1] (x; ε)) (x 2 − M[≤n0 +1] (x; ε))−1 ,
(6.3)
... g
[n] def
=
Xn0 −1 (x) Yn−1 (x; ε) ψn ([n] (x; ε)) (x 2 − M[≤n] (x; ε))−1 ,
and so on, using indefinitely the identity 1 ≡ ψn ([n] (x; ε))+χn ([n] (x; ε)) to generate the new propagators. In this way we obtain a formal representation of h as a sum of tree values in which only renormalized trees (i.e. with neither trivial nodes nor self-energy clusters, see Definition 4 in Sect. 4 and in which each line carries a scale label [n ]. This means that we can formally write h as in (5.10), with Val(θ ) defined according to (5.11), but now the scale label [n ] is such that n can assume all integer values ≥ −1, and no line carries a scale label like [≥ n]: only scale labels like [n] are possible. We can summarize the discussion above in the following definition.
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Definition 7. Given a sequence M[≤m] (x; ε) as in Definition 6, m ≥ 1, let M[n] (x; ε) = M[≤n] (x; ε) − M[≤n−1] (x; ε) with M[≤0] ≡ M[0] ≡ M0 so that M[≤n] (x; ε) = n [m] (x; ε). Setting [n] (x; ε) ≡ D(x) if n ≤ n , define for all n ≥ 0, 0 m=0 M [m] (x; ε)) ψn ([n] (x; ε)) n−1 m≥0 χm ( g [n] (x; ε) = (6.4) x 2 − M[≤n] (x; ε) (for n = 0 this means ψ0 (D(x)) (x 2 − M0 )−1 ). We say that g[n] = g [n] (ω · ν ; ε) is a propagator with scale [n]. The matrices M[m] (x; ε) will be defined as in Sect. 5 for n ≤ n0 and will be defined recursively also for n > n0 in terms of the self-energy clusters R Sk,n−1 introduced in Definition 4, Sect. 5, setting for n > n0 (see (5.9)) M (x; ε) = [n]
n−1 m=0
χm (
[m]
∞ (x; ε))
VT (x; ε),
(6.5)
k=2 T ∈S R
k,n−1
where the self-energy values VT (x; ε) are evaluated by means of propagators on scales less than [n]. Note that we have already defined (consistently with (6.5)) the matrices M[≤n] with n ≤ n0 and the propagators on scale [−1], [0], . . . , [n0 − 1] (so that (6.4) defines also g [n0 ] (x; ε)). Remark. (1) Some propagators may vanish being proportional to a product of cut-off functions. If a propagator of a line has scale [n] and does not vanish then, see (6.4), 2−2(n+1) C02 ≤ [n] (x; ε).
(6.6)
Note that for n < n0 a similar upper bound holds because of the independence of [n] from n. We shall see that this happens also for n ≥ n0 because the eigenvalues do not move too much along the iterative scheme (see Remark (3) to the inductive assumption below). (2) Our definitions of the matrices M[≤n] (x; ε) for n > n0 will be such that given the node harmonics of a tree the scale [n] that is attributed to a line can only assume up to two consecutive values unless the propagator (hence the tree value) vanishes, see Remark (3) to the inductive assumption below. (3) We may and shall imagine that scale labels are assigned arbitrarily to each line of a given tree with the constraint that no self energy clusters are generated; however the tree will have a non-zero value only if the scale labels are such that all propagators do not vanish. This means that only up to two consecutive scale labels can be assigned to each line if the tree value is not zero. The “ambiguity” on the scale labels for a line is due to the use of the non-sharp χ and ψ functions of Fig. 2. We make an inductive assumption on the propagators on the scales [m], 0 ≤ m < n. Inductive Assumption. Let n0 ≡ n0 + n (see (4.2)) and suppose n0 large enough; then (i) For 0 ≤ m ≤ n − 1 the matrices M[m] (x; ε) are defined by convergent series for all ε ∈ IC and, for all x, they are Hermitian, and M[m] (x; ε) = (M[m] (−x, ε))T . Furthermore they satisfy the same relations as (5.13), hence (5.14), with n replaced by m, for all 0 < m < n − 1, with suitably chosen (new, possibly different) constants κ1 , A, A , B, τ . One can take τ = 2τ1 . o , m = 0, . . . , n, with E o ⊂ I , such that, defin(ii) There exist K > 0 and open sets Em C m [m] [m−1] (ε) for m = n0 , . . . , n − 1 by (i) in Definition 6 ing recursively λj (ε) in terms of λj
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G. Gentile, G. Gallavotti def
[0] above, while setting λ[m] j (ε) ≡ λj for m = 0, . . . , n0 −1, and defining τ1 = τ0 +r +1, o see (3.4), one has for ε ∈ Em and all independent choices of the signs ±,
[m] [m] ± (ε) , min λ (ε) ± λ (ε) [m] (x; ε) = min min x ± λ[m] x j j i j
≥2
− 21 m
j ≥i
C0 , |ν|τ1
1
o |Em | ≤ K2− 2 m C 2 ,
(6.7)
for all m ≤ n − 1 and all x. Remarks. Assuming validity of the hypothesis for m < n we note a few of its implications. (1) So far we have only checked the hypothesis for scales [m] with m ≤ n0 , as expressed by Lemma 2 in Sect. 5, i.e. for the high frequency propagators. If (i) is proved also for o of ε’s of measure m = n then we can impose (6.7) immediately by excluding a set Em 1 estimated by 2− 2 m C 2 K with K a constant that can be bounded in terms of A , A by as in (A2.1), with λ[0] (ε) replaced by λ[m] (ε), and introducing the constants ρm and ρm j j proceeding as is done in Appendix A2 for the case n ≤ n0 . Note that since the self[0] energies λ[m] j (ε) are ≡ λj (ε) for all m = 0, . . . , n0 − 1 one will have, for such m’s, o ≡ I /E Em C n0 −1 , see (3.3). It is very important to keep in mind, in the above argument, that the self-energies either are 0 (for j ≤ r) or are close within O(ε 2 ) to the positive eigenvalues of M0 , and they are differentiable in ε and to the right and left of each x by (i); see (5.14). (2) To exploit the cancellations discussed below (and with more details in Appendix A3.3) we shall have to consider also trees whose value is zero as they contain lines with propagator which is vanishing because of the χ , ψ cut-off functions in the definition (6.4). Nevertheless we shall see (next remark) also that if a line with a scale [n] has vanishing propagator (i.e. g [n] (x; ε) = 0) then n differs at most by one unit from the integer n such that g [n ] (x; ε) = 0. Thus if we consider [n] (x, ε) we can bound it by changing the inequality (6.6) into C02 2−2(n+2) < [n] (x; ε). (3) By (5.13) and (5.14) and (I) in Appendix A4, we deduce that λ[m] j (x; ε), hence n/(2τ1 ) [m] 2 −κ 2 1 λj (ε), do not change by more than C B ε , with respect to λ[0] n≥n0 e j (ε), [n] [n+1] if ε < ε 1 (and m ≥ n0 ). And for n ≥ n0 the quantities (x) and (x) differ by a n/τ quantity bounded by ε2 e−κ1 2 which is extremely small compared to C02 2−2n so that, using also the characteristic functions in (6.4), we deduce the property in Remark (2) following Definition 7 essentially for the same reasons why the corresponding property held in the cases n < n0 (where [n] (x) is n–independent). (4) Hence if ε is small enough the self-energies, i.e. λ[m] j (ε), have distance bounded above by 2as ε and below by 21 ε min a1 , minj {aj +1 − a1 } = 2ρ εas with ρ defined in (4.2), if ε is small enough, say ε < ε 2 . (5) Therefore by Remark (4) we see that the distance of |x|2 from the closest value 2 λ[m] j (ε) is smaller than one fourth, up to corrections O(ε ), the distance between the 2 −2m < ρεa distinct values of λ[m] s j (ε), if m is large enough compared to n0 , i.e. if 2C0 2 (or m − n0 ≥ n as implied by the definition (4.2) of n). This means that j (x) is
Degenerate Elliptic Resonances
341
m, ε-independent and it coincides with the label minimizing x 2 − |λ[m] j (x; ε)| for all m ≥ n0 and all ε ∈ I . 0 −1] (ε) ≡ λ[0] (6) λ[n j j are x-independent and, by their definition, the same remains true [m] for all λ[m] j (ε). The self-energy λj (ε) will be thought of as a reference position for the j th eigenvalue on scale [m], m ≤ n − 1. (7) As noted in Remark (5) the quantity x 2 − λ[n] j (x) (x; ε) is the smallest denominator appearing in the value of the propagator of a line with momentum ν if g [n] (x; ε) = 0 (here x = ω · ν). The key to the analysis is the check that the quantities [n] (x; ε) can be used to bound below the denominators of the non-vanishing propagators of scale [n]. If [n] 2 2 λ[n] j (x) (x; ε) < 0 one has x − λj (x) (x; ε) ≥ x , so that the assertion is trivially satisfied:
therefore the really interesting case is when λ[n] j (x) (x; ε) ≥ 0 (which includes the cases j (x) > r). If x has scale [n] with n ≥ n0 one has 2 [n] [n] [n] [n] x − λj (x) (x; ε) ≥ x 2 − λj (x) (ε) − λj (x) (ε) − λj (x) (x; ε)
1 [n] [n−1] [n] −(n+3) ≥ x 2 − λ[n] (ε) + 2 C − λj (x) λj (x) (ε), ε − λj (x) (x; ε) (6.8) 0 j (x) 2 1 1 ≥ x 2 − λ[n] (ε) ⇒ x 2 − M[n] (x, ε) ≥ x 2 − λ[n] (ε) , j (x) j (x) 2 2 having used the lower cut-off ψn ([n] (x; ε)) in the propagator (see (6.3)) to obtain the first two terms in the second line (and added a further factor 2−1 in order to extend the result also to the propagators considered in Remark (2)), while the upper cut-off χn−1 ([n−1] (x; ε)) has been used to obtain positivity of the difference between the second and third terms in the second line, after applying (5.14), for n ≥ n0 , to get 2 max |∂x± λ[n] j (x) (x; ε)| ≤ B ε , j (x) > r, x
2 2 −2n |λ[n] , j (x) (x; ε)| ≤ B ε |x| ≤ ε C0 2
j (x) ≤ r,
(6.9)
so that the last term in the second line of (6.8) can be bounded above for some B, proportionally to ε2−n C0 . Hence the first inequality in the last line of (6.8) follows if ε small enough, say ε ≤ ε 3 for some ε 3 , fixed independently of n. The latter constraint can be achieved simply by taking n0 large enough, see (3.2). The last implication if j (x) > r and follows from (6.9) if j (x) ≤ r because λ[n] j (x) (ε) = 0. Otherwise
√ √ [n] [n] 1 |x|, λ[n] j (x) (ε), λj (x) (x, ε) ∈ [ 2 ε a1 , 2 as ε] one has (|x| + λj (x) (x, ε))/(|x| +
−2 √a /a , as long as ε < ε (see Remark (4) above): implying again λ[n] 1 s 2 j (x) (ε)) ≥ 2 (6.8). Hence [n] (x; ε) can be effectively used to estimate the size of the non-vanishing propagators which is, therefore, closely related to the scale of the corresponding lines. (8) The Diophantine condition (3.3) and (6.7) will play from now on a key role. We begin by remarking that if the inductive hypothesis is proved all lines will eventually acquire a [n] −2n 2 well defined scale label: in fact
for fixed x one cannot have (x, ε) ≤ 2 C0 for all n
−n because this implies5 ||x|− λ[n] j (x) (ε)| < 2 C0 , which sooner or later becomes incompatible with the first of (6.7). This explains why there is no trace left of the propagators g [≥n] (x, ε). 5
As |a 2 − b2 | < c2 implies |a − b| < c for a, b, c > 0.
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To estimate the corrections to the self-energy as n increases it is clear that we must estimate the size of M[n] (x; ε). For this purpose we need the following result. Lemma 3. There is ε small and constants κ1 , A, A , B such that if ε < ε and the inductive hypothesis is assumed for 0 ≤ m ≤ n − 1 then the matrix M[n] (x; ε) can be bounded by (5.13) and the inductive hypothesis holds for m = n. Hence the hypothesis holds for all n since we have already checked it for n = 0, . . . , n0 (Lemma 2). The new constants κ1 , A, A , B will be different from the ones determined in Lemma 2. They will be n-independent, a property checked by a word by word repetition of the corresponding argument in Appendix A3. Proof. For n ≤ n0 the bound (5.13) is covered by Lemma 2. So we can assume n ≥ o n0 + 1. Suppose first ε ∈ ∩n−1 m=n0 −1 Em , with Em = IC \ Em , so that the Diophantine R property (6.7) holds for all m ≤ n − 1. Consider a self-energy cluster T in ∪∞ k=2 Sk,n−1 . If the entering and exiting lines (with propagators of scale [≥ n]) have momenta ν we begin by showing that |ν v | > 2(n−6)/(2τ1 ) . (6.10) v∈V (T )
Indeed the cluster contains at least one line = v with propagator which we can suppose to be not vanishing and which has scale [n − 1]. We can write ν = ν 0 + σ ν, where σ = 0, 1 and we set ω · ν = x, ν 0 = w∈V (T ) ν w , and finally x = ω · ν . w v
Since the line is not on scale [n − 2] (as it is on scale [n − 1]) it follows from (6.3) that
|x | − λ[n−2] (ε) ≤ 2−(n−2) C0 . (6.11) j (x ) Therefore if (6.10) does not hold and
if σ = 0, by the first part of the Diophantine conditions (6.7), one finds |x | − λ[m] (ε) > C0 2−m/2 2−(n−6)/2 for all m ≤ n − 1 i
and for all 1 ≤ i ≤ d, which would be in contradiction with (6.11). If instead σ = 1 we shall use the second part of the Diophantine conditions (6.7) and get a contradiction. Remark that x can be assumed to be on scale [q] with q ≥ n [p] because of the cut-off functions in (6.5) so that one has |x| − λ (ε) ≤ C0 2−p for j (x)
p ≤ n − 1. Hence if x satisfies (6.11) we get, by assuming that (6.10) does not hold,
[n−2] 23−n C0 ≥ |x | − λ[n−2] (ε) + − λ (ε) |x| j (x ) j (x)
[n−2] ≥ x − x + η λ[n−2] (6.12) j (x ) (ε) + η λj (x) (ε) ≥
C0 (n−2)/2 2 |ν
− ν|τ1
=
C0 (n−2)/2 2 |ν 0 |τ1
≥ 24−n C0 ,
for some η, η = ±1, which again leads to a contradiction, so that (6.10) holds also in such a case. Every node factor contributes to M[n] a factor fν v bounded by F0 e−κ0 |ν v | ; there are ≤ (4d 2 )k k! self-energy clusters, 4k scales (for each line there are only two scales for which the propagator is not zero, and one has to allow also a scale different by one
Degenerate Elliptic Resonances
343
unit from that which corresponds to have a nonvanishing propagator, see Remark (3) after the inductive assumption), and Nm (T ) lines of scale m = −1, 0, 1, . . . , n in each self-energy cluster T contributing to M [n] (x; ε) and not to the M [m] (x; ε), with m < n. Thus the bound on the graphs contributing to M [n] (x; ε) and with no lines of scale [−1] is G0
∞ k=2
ε
k
1 Gk1 e− 2 κ0
v∈V (T )
|ν v | −G2 2n/(2τ1 )
e
n
22mNm (T ) ,
(6.13)
m=0
for suitable constants G0 , G1 , G2 , explicitly computable by the above remarks. The estimate of the number N m (T ) is given in Appendix A3 (cf. in particular Sect. A3.4), and gives Nm (T ) ≤ Em v∈V (T ) |ν v |, with Em = 2(6−m)/(2τ1 ) and τ1 in (3.4), which shows convergence of the series in (6.13) if ε is small enough, say ε < ε. Considering also the graphs with lines of scale [−1] makes the estimates worse by a 1 factor ε− 2 k because the number of lines with scale [−1] cannot exceed 21 k and somewhat increases the constant Gj : their propagators are bounded below by a constant times ε; however their number cannot be larger than 21 k in trees of order k (see Remark (5) in Sect. 2). Therefore they may reduce the factor εk normally present in the value of a graph 1 with k nodes to ε 2 k ; hence this will not affect the convergence of the series other than by putting a more severe constant on the maximum value of ε. For k < 4 the exponent of ε can be replaced by 2, see Remark after Definition 4. We can and shall assume that ε does not exceed min{ε1 , ε2 , ε3 }, with ε 1 , ε 2 and ε 3 introduced earlier (see Remarks (3), (4), (7) after the inductive hypothesis). The rest of the argument repeats the analysis in Appendix A3 with minor notational changes: we only hint at the details in Appendix A3.4. Under the considered hypotheses the matrices M[n] (x; ε) are well defined, by the above discussion on convergence of the defining series on the set ∩n−1 m=n0 −1 Em . The symmetry in item (i) is due to algebraic identities valid for the Lindstedt series. They are detailed in Ref. [GG], Appendix A5, for ε < 0: being of algebraic nature the argument does not depend on the sign of ε and it holds unchanged in the present case. The second and third lines of inequalities in (5.13) embody the cancellations. We need to check the cancellations, to make sure for instance that the structure of the matrix M[n] (x; ε) preserves the eigenvalues, and the Whitney smoothness: a danger being that the first r eigenvalues become “detached” from 0, i.e. no longer can be bounded by εx 2 , see (5.14). For instance a bound like O(ε2 ) would not be enough as it would imply that the self-energies λ[n] j (ε) may become different from zero for j ≤ r. Since the function M[n] (x; ε) is defined on the complement of a dense open set, differentiability in the sense of Whitney can be proved (as usual) by computing a formal derivative and then showing that it is continuous and that it can also be used as a bound in interpolations.6 The computation of the formal derivatives proceeds as the computation of the actual derivatives done in the proof of Lemma 2 (in Appendix A3). One proves formal 6 More precisely in its simplest form Whitney’s theorem states that if F (x) is a function defined on a closed set C of the interval [0, 1] and if there is a continuously function F (x) defined on C and if for some γ > 0 and all x, y ∈ C one has |F (y) − F (x)(y − x)| < γ |x − y| (we call this an interpolation bound) then there is a continuously differentiable function F (x) extending F to [0, 1] and with derivative F (x), with max |F (x)| < γ , extending F (x).
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right and left continuous differentiability of the matrices M[n] (x; ε) on the closed set ∩n−1 m=n0 −1 Em simply by differentiating term by term the value of each cluster contributing to M[n] (x; ε). This involves differentiating matrices like (x 2 − M[≤p] (x; ε))−1 , i.e. the matrices M[p] (x; ε) with p < n, which are differentiable by the inductive assumption, [p] or it involves differentiating the cut-off functions ψp , χp and the locations λj (ε) with [p]
j > r (because λj (ε) ≡ 0 for j ≤ r) which appear in the form [p] (x, ε) in the arguments of the cut-off functions. All such quantities are differentiable in ε and right and left differentiable in x by the inductive assumption; furthermore all terms arising [p] from differentiation either of M[p] (x; ε) or λj (ε), with p < n, appear multiplied by some power of ε, so that the inductive assumption is found to hold also for p = n (for a similar discussion see Ref. [Ge]). Note that [n] (x; ε) depend on j (x) but as ε varies within the interval I , see (ii) in Definition 1, j (x) is not only ε-independent but it is also constant in x for x varying in small intervals near the eigenvalues of M0 and, therefore, in intervals widely spaced because n ≥ n0 : this is due to the cut-off functions which force x to be close to a single eigenvalue if the propagator of the corresponding line is different from 0. Hence for n ≥ n0 we do not have to differentiate the function j (x) (neither with respect to x nor with respect to ε from which it does not depend); for n < n0 the function j (x) is constant to the right and to the left of every point. The n-independence of the constants A , A, B appearing in the inductive hypothesis is proved word by word as the corresponding argument in Appendix A3; the constant κ1 has been estimated above (see G2 in (6.13)) and is n-independent. The interpolation bound, see the footnote below, necessary for defining the Withney derivatives, holds because in comparing two contributions to M[n] (x; ε) with different x or different ε the difficulty might only come from the comparison of (x2 − M[≤p] (x , ε))−1 evaluated at two different points and for one line at a time: this can be done algebraically by using the resolvent identity −1 −1 −1 2 x2 − M[≤p] (x , ε) − x − M[≤p] (x , ε ) = x2 − M[≤p] (x , ε) · −1 2 2 , (6.14) · x − x2 + M[≤p] (x , ε ) − M[≤p] (x , ε) x − M[≤p] (x , ε ) which involves only denominators evaluated at x, ε’s which are in the set where they are controlled by the (6.7) and therefore can be estimated in the same way as the formal derivatives. The Whitney extension is therefore possible keeping control of the bounds for all ε’s (small as above) and x. The dependence on x may involve the functions D(x) (for p ≤ n0 − 1) so that the differentiability in x will be possible only to the right and to the left of each point (this involves a natural generalization of Whitney’s theorem). The cancellations analysis (i.e. the proof of the second and third inequalities in (5.13)) is inductive and has been performed several times in the literature, see Refs. [Ga2, GG]. In Appendix A3 we have repeated it following the version in Ref. [GM1] with some minor modifications. The same proof applies to the present case (being a purely algebraic check). The inequalities (5.13) imply (5.14) and therefore we get differentiability of the matrices M[≤n] (x; ε) and of the self-energies. This allows us to impose validity of (6.7) by excluding a few more values of ε by Remark (1) to the inductive hypothesis. Therefore we conclude that M[n] (x; ε) is defined and verifies (5.13) (with suitably chosen constants κ1 , A , A, B) in the same domain ε < ε, where the matrix M[≤p] (x; ε)
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345
is already defined for p ≤ n − 1. Of course M[n] will be relevant for our analysis only on the set ∩nm=n0 −1 Em and the extension outside such set is only useful to simplify the analysis as it allows us to use freely interpolation formulae, mainly to check (6.7). The matrix M[≤n−1] (x; ε) verifies the inductive assumption although it has physical meaning only for ε ∈ ∩∞ m=n0 −1 En , where En is the domain in which (6.7) holds for m ≤ n. Having checked that the series defining the M[≤n] (x; ε), hence the self-energies, converge and verify the bounds in the inductive hypothesis we still have to check that the fully renormalized series for h, which has thus been shown to make sense term by term, converges and that its sum is actually a function h satisfying the equations for the parametric representation of invariant tori. To study convergence we can take again advantage of the method, already used in the proof of Lemmas 2 and 3 above to estimate the number of lines on scale n in a self-energy cluster containing no self-energy clusters. Indeed also for renormalized trees one can prove a bound like Nm (θ ) ≤ Em v∈V (θ) |ν v | for the number Nm (θ ) of lines on scales def
m contained in (θ) with Em fast decreasing with m: Em = 2(6−m)/(2τ1 ) (see Appendix A3). Hence convergence in the region E ∈ ∩∞ n=n0 −1 En follows because if we only sum values of trees without self-energy clusters then we can use the above bound on Nm (θ ). The set En0 , complement of En in IC , has measure estimated by C 2 2−n/2 K for ε ∈ (( 21 C)2 , C 2 ] = IC . Since C = 2−n0 C0 and n ≥ n0 − 1 > n0 this is a very small fraction of the interval IC and the smaller the closer is IC to 0. This means that the set of ε’s for which the whole construction can be performed has 0 as a density point. Note that the resummation just defined is a real resummation of our series only for ε ∈ ∩∞ n=n0 −1 En , and there it gives a well defined function. The check that the functions h(ψ) defined by the convergent renormalized series evaluated at ψ = ωt do actually solve the equations of motion can be performed by repeating the corresponding analysis in Ref. [Ge]. The equation that h = (a, b) has to solve is h = ε g (∂α f (ψ +a, β 0 +b), ∂β f (ψ +a, β 0 +b)), where g is the pseudo-differential operator (ω · ν)−2 . The proof is of algebraic nature and ultimately follows from the fact that the series we are considering is a resummation of Lindstedt’s series which is a formal solution of the problem. This explains why the various algebraic identities necessary for the check actually hold and the proof proceeds exactly as in Sect. 8 of Ref. [Ge]: we reproduce the argument and the chain of identities in Appendix A5. Therefore the proof of Theorem 1 in Sect. 1 is complete. 7. Concluding Remarks The analysis can be immediately extended to the case in which the matrix ∂β2 f0 (β 0 ) has some non-degenerate positive eigenvalues and some additional negative ones. The negative eigenvalues give no problems and they can be treated as in the case of Ref. [GG] in which all eigenvalues are negative. The negative eigenvalues do not give rise to new small divisors, unlike the positive ones; in more physical language the proper time scales (i.e. real proper frequencies) of the tori cannot resonate with the time scales of hyperbolic type (i.e. imaginary) introduced by the perturbation. Hence the following generalization of Theorem 1 holds. Theorem 2. If the matrix ∂β2 f0 (β 0 ) is not singular and has pairwise distinct eigenvalues the conclusions (i), (ii) and (iii) of Theorem 1 in Sect. 1 follow also in this case.
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The present work has developed a combinatorial approach to the proof that the frequencies of elliptic type possibly introduced by the perturbation do not resonate with the proper frequencies of the tori at least if ε is not too special in a small interval [0, ε], i.e. if it is in a set E ⊂ [0, ε] of large measure near 0. Nevertheless the complement of E is an open dense set in [0, ε]. The results hold for the Hamiltonian (1.2) and the special resonances (ω, 0) considered: they can be extended to the most general resonances of Hamiltonians like (1.2) with a general quadratic form for the kinetic part (i.e. with 21 I · I replaced by 21 I · QI with Q a non-degenerate d × d matrix). The case of ∂β2 f0 (β 0 ) with degenerate eigenvalues seems quite different from the one treated here. Degeneracy will be removed to order O(ε 2 ) under generic conditions. However O(ε2 ) is also the order of variation of the self-energies and one has to find a way to perform the resummations even between scale n0 and scale 2n0 , which is the scale at which the singularities of the propagator are split apart and one shall be able to proceed in the same way as we did in the case of non-degenerate eigenvalues. The Lipschitz regularity in ε in Theorems 1 and 2 can be replaced by C k regularity for any k by exploiting the comments in Remark (2) to Lemma 2 and Remark (2) in Appendix A3.2. Unfortunately there seems to be no example known in which one can check that the power series studied here are divergent as power series in ε. Note that the (infinitely many) divergent series that have arisen in this paper are obtained by first splitting the coefficient of order k in the Lindstedt power series and then collecting contributions from the different orders in ε: the latter form divergent series for which we have assigned a summation rule. Therefore we have not proved divergence of the Lindstedt series as power series in ε: in this sense (unlikely) convergence of the Lindstedt series has not been ruled out (yet). Nor is there any uniqueness result on the value of the renormalized series. The latter depends on quite a few arbitrary choices (even in the hyperbolic cases); for instance the cut-off shapes in Fig. 2 are quite arbitrary and in principle the allowed ε’s will change with the choice. Furthermore, although we have not really checked all necessary details, it seems to us that our method also shows that, given a value ε0 for which the renormalized series converges, one can find a complex domain of ε which is open, reaches the real axis with a vertical cusp at ε0 and extends to an open region including a segment (−η, 0) on the negative real axis. In this domain the renormalized series should converge taking on the real axis real values parameterizing an hyperbolic torus with the same rotation vector. However since there are no uniqueness proofs we cannot guarantee that each such extension does not correspond to a different torus (close within any power of ε to any other torus of the same type as ε → 0)). This would signal a “giant bifurcation” that one would like to exclude; in Ref. [GG] an attempt was made to show uniqueness by estimating the size of the Lindstedt series coefficients aiming at applying the theory of Borel transforms. However we could not prove good enough bounds. We obtained k!α growth with a too large α (given our estimated size of the domain of analyticity in ε) to apply uniqueness results from the theory of Borel summations.
Appendix A1. A Brief Review of Earlier Results The system which is usually studied in literature when the problem of persistence of lower-dimensional elliptic tori is studied, is of the form
Degenerate Elliptic Resonances
H = ω(ξ ) · A +
347 s
k (ξ ) qk2 + pk2 + P (α, A, q, p),
(A1.1)
k=1
where (α, A, p, q) ∈ Tr × Rr × Rs × Rs . The function P is analytic in its arguments, and ξ is a parameter in Rr ; the function P is a perturbation: this means that a rescaling of the actions could allow us to introduce a small parameter ε in front of the function P . The frequencies of the harmonic oscillators are called normal frequencies; the case k (ξ ) = k = constant (that is with the normal frequencies independent of ξ ) is a particular case, and it is usually referred to as the “constant frequency case”. Existence of invariant tori for the system (A1.1) was originally proved by Mel’nikov [Me1, Me2], new proofs were produced by Kuksin [Ku1], Eliasson [E1], and P¨oschel [P1]. The case s = 1 is easier, and it was earlier solved by Moser [Mo]. Later proofs were given by R¨ussmann, see for instance Ref. [R]. See also the very recent Ref. [LW]. For P = 0 the dimension of the tori is r < d and the variables (q, p) move around stable equilibrium points, hence such tori are called elliptic lower-dimensional tori. The conditions under which the quoted results are proved are, besides the usual Diophantine condition (1.3) on ω, two non-resonance conditions involving one and two normal frequencies (the so-called first and second Mel’nikov conditions, originally introduced in Ref. [Me1]); in particular one has to impose that the normal frequencies are non-degenerate (i.e. they have to be all different from each other). Recently proofs of existence of elliptic lower-dimensional tori were given by requesting only the first Mel’nikov conditions: this allows treating degenerate frequencies. The first result in this direction is due to Bourgain [Bo3], where the ideas introduced in Refs. [CrW, Bo1] to prove existence of periodic and quasi-periodic solutions in nearly integrable Hamiltonian partial differential equations were adapted to construct lowerdimensional tori in the finite-dimensional Hamiltonian systems (A1.1) corresponding to the case of constant normal frequencies. New proofs, extending the results also to the case of non-constant normal frequencies, are due to Xu and You [Y, XY]. An extension of the results of existence of periodic and quasi-periodic solutions describing lower-dimensional invariant tori for infinite-dimensional PDE systems has been provided in a series of papers, which include Refs. [Ku1, Ku2, Wa, CrW, KP, P2, Bo1, Bo2, Bo4, BKS, GM2, GMP]. On the other hand the problem (1.2) has not been widely studied in literature. It corresponds to a degenerate case because in absence of perturbations the lower-dimensional tori are neither elliptic nor hyperbolic: it is the perturbation itself which determines if the tori, when continuing to exist, become elliptic or hyperbolic (or of mixed type or parabolic). (i) The case of hyperbolic tori is easier, and it was the first to be studied, by Treshch¨ev [T]. Recently the problem was reconsidered in Ref. [GG], where the analyticity domain of the invariant tori was studied in more detail. In the case of elliptic tori the problem was considered in Refs. [ChW, WC], where Treshch¨ev’s approach to the study of the case of hyperbolic tori, involving a preliminary change of coordinates, is used to cast the Hamiltonian in a form which is suitable for applying P¨oschel’s results on elliptic tori: in particular this imposes the same conditions as in Ref. [P1] on the normal frequencies which appear after the canonical change of coordinates is performed. (ii) The existence problem has been also considered in Ref. [JLZ], where elliptic and hyperbolic tori were studied simultaneously, again by imposing some non-degeneracy conditions on normal frequencies. Ref. [JLZ] does not investigate resummations of Lindstedt’s series; it is based on a rapid convergence method, close in spirit to the
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original proofs of the KAM theorem: a concise existence proof of lower-dimensional tori is achieved in both the elliptic and hyperbolic cases. We do not know whether the tori whose existence is proved in this alternative way coincide with the ones constructed here: this is due to the lack of analyticity at ε = 0 and the consequent lack of a uniqueness proof, see the last comment in Sect. 7. In our opinion the problem of the identity of the tori that we have studied here and in Ref. [GG] with those previously studied in Refs. [T, JLZ] is an open and important problem on the subject. We stress that in all quoted papers, except Ref. [JLZ, T], the problem is considered with ε (i.e. the size of the perturbation) fixed and the study deals with estimates of the measure of the rotation vectors ω for which there exist invariant tori. We suppose, instead, that ω is fixed, hence we study the dependence on ε of the lower-dimensional invariant tori and, in particular, the set of values of ε for which the tori survive. Our techniques extend those in Refs. [GG, Ge], and are based on the method introduced in Refs. [E2, Ga2]. With respect to Ref. [Ge], where existence of quasi-periodic solutions is proved for the generalized Riccati equation considered in Ref. [Ba], the main difficulty is due to the presence of several normal frequencies. It is not surprising that this generates extra technical difficulties: as already noted, it is well known that the case s = 1 is easier; see Refs. [Mo, Ch2]. An advantage of the present method is that it is fully constructive and gives a very detailed knowledge of the solution. Appendix A2. Excluded Values of ε Define
ε [0] [0] min min |∂ε λ[0] (ε)|, min |∂ λ (ε) − ∂ λ (ε)| , ε ε i j i i=j i>r as i,j >r 1 def max λ[0] (ε) , (A2.1) ρn 0 −1 = √ j εas j def
ρn0 −1 =
and note that ρn0 −1 is bounded from below proportionally to ρ, as defined in (4.2), and ρn 0 −1 = 1. Then (3.3) excludes, for each ν, an interval in ε whose measure is bounded √ (using as ε ≤ C; see (3.2)) by 2−(n0 −1)/2 C C0 K0 |ν|−τ1 ,
(A2.2)
where the constant K0 can be estimated by K0 = s as−1 ρn−1 . 0 −1 The Diophantine condition on ω implies that if (3.3) is invalid then |ν| cannot be too small
2 εas ρn 0 −1 + 2−(n0 −1)/2 C0 |ν|−τ1 ≥ |x| ≥ C0 |ν|−τ0 . (A2.3)
Therefore εas ρn 0 −1 ≥ 41 C0 |ν|−τ0 if n0 ≥ 3, hence in this case we only have to consider
√ the values of ν with |ν| ≥ (C0 /(4 εas ρn 0 −1 ))1/τ0 . Since C/2 < εas ≤ C = 2−n0 C0 ,
we get the bound (3.5) with τ1 = τ +r +1 and K = K0 C0 (4C ρn 0 −1 C0−1 )(τ1 −r−1)/τ0 1 1 ν=0 |ν|r+1 = 4K0 ρ n0 −1 ν=0 |ν|r+1 . Note that a condition like τ1 > τ + r is sufficient to obtain both summability over ν and a measure (of the excluded set) relatively small with respect to that of IC . If n0 < 3, hence n0 < 3, the same conclusion trivially holds possibly increasing the value of K by a factor 4.
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Appendix A3. Resummations: Convergence and Smoothness To prove Lemma 2, we first show that the series defining M [n] (x; ε) for 0 ≤ n ≤ n0 converge and then we check smoothness and the bounds. This is done for completeness as the argument is almost a word by word repetition of the analysis in Ref. [GG], with a few slight changes of notations necessary to adapt it to our present notations and scope. To study convergence of the series defining M [n] (x, ε), n ≤ n0 , we remark that we have to consider only trees in which all propagators have scales [p] with p ≤ n0 . Therefore the propagators which do not vanish will be such that their denominators satisfy D(x) > 2−2(n+1) |x|2 , see (4.4), so that they are effectively estimated from below by |x|2 times a constant. Note that the case n = 0 is obvious (and it is treated in Sect. 3). A3.1. Convergence. We suppose that the eigenvalues of M[≤p] (x; ε), n = 0, . . . , n−1, [p] differ from the corresponding ones of M[≤0] (x; ε) ≡ M0 so that |λj (x, ε) − λ[0] j | < 1 2 2 −2n−2 γ ε for some γ > 0, and that ε is small enough so that γ ε < 2 εas 2 and, there[p] fore (see (4.4)), if a line with frequency x has scale [p], p < n, then |x 2 − λj (x, ε)| > 2−2(n+2) x 2 . We shall use that if the propagator of a line is on a scale [n] then one has D(x) ≤ 2−2(n−2) C02 , even though we could allow also a bound D(x) ≤ 2−2(n−1) C02 . The reason for this is again for later use in bounds necessary to establish the needed cancellations as commented in Sect. A3.2. R Consider a renormalized self-energy cluster T ∈ Sk,n−1 , and define m (T ) = { ∈ (T ) : n = m}, for m ≤ n − 1, and P(T ) the set of lines (path) connecting the external lines of T . If ν is the momentum flowing in the line entering T then the momentum flowing in a line ∈ (T ) of scale [p], p ≤ n − 1, will be ν 0 + σ ν, σ = 0, 1, where ν 0 is the momentum that would flow on if ν = 0. The corresponding frequency will be x = x0 + σ x, with obvious notations. First of all we shall prove the bound |ν v | ≥ 2(n−n−5)/τ0 (A3.1) v∈V (T ) R for T ∈ Sk,n−1 . If there is a line ∈ n−1 (T ) which does not belong to P(T ) then 0 x = x , so that (A3.1) follows from the Diophantine condition on ω. If all lines in n−1 (T ) belong to P(T ), consider the one among them, say , which is closest to 2T , i.e. the entering line of T . Then call T1 the connected set of nodes and lines between7 and 2T . If T1 is a single node v then ν v = 0, otherwise v would be a trivial node; if T1 is not a single node then by construction all the lines of T1 have scales strictly smaller than n, hence x = x otherwise T1 would be a self-energy cluster. In both cases one has |x − x| = |x0 | > C0 | v∈V (T1 ) ν v |−τ0 . On the other hand both D(x) and D(x ) must be ≤ (C0 2−(n−2)+1 )2 hence, by (4.4) |x|, |x | ≤ C0 2−n+n+3 , so that |x − x | ≤ C0 2−n+n+4 , and (A3.1) follows also in such a case. The next task will be to show that the number Nm (T ) of lines on scale [m], with m ≤ n − 1, contained in a cluster T is bounded by Nm (T ) ≤ max{Em v∈V (T ) |ν v | − 1, 0}, 7 The lines between two lines and with < are all the lines which precede but which do 1 2 2 1 1 not precede 2 nor coincide with it.
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with Em = E 2−m/τ0 for a suitably chosen constant E; as it will emerge from the proof one can take E = 2 2(n+4)/τ0 . Before considering clusters we adapt to our context the classical bound (Siegel-Bryuno-P¨oshel; see also Ref. [Ga2] and references quoted therein), stating that, if Nm (θ ) denotes the number of lines on scales [m], then by induction on the number of nodes of θ one shows: Nm (θ ) ≤ max{Em v∈V (θ) |ν v | − 1, 0}. Indeed if θ contains only one node v0 and the frequency x = ω · ν v0 of the root line has scale [m] one has 2−m+1 C0 ≥ D(x) ≥ 2−(n+1) |x| ≥ 2−(n+1) C0 |ν v0 |−τ0 ⇒ |ν v0 | > 2(m−n−2)/τ0 , (A3.2) hence Em |ν v0 | − 1 ≥ 2 and the bound holds in this simple case. If θ has k nodes and the root line does not have scale [m] the inductive assumption, if it is assumed for the cases of k < k nodes, gives the bound for k-nodes trees. If the root line has scale [m] then on each path of tree lines leading to the root we select the line among the ones on scales [m ] with m ≥ m closest to the root (if any is found on the path) and we call the selected lines 1 , . . . , q . If q = 1 either the bound follows just as in the case of k = 1 (when q = 0) or from the inductive hypothesis (when q ≥ 2). The case q = 1 and [n1 ] = [−1] (i.e. ν 1 = 0) can be treated as the case q = 0. If q = 1 and ν 1 = 0, by construction all lines between the root line and 1 , see the footnote 7, have scales [m ], with m < m, so that such lines, together with the nodes they connect, form a cluster T . The frequencies x and x1 must be √ different because the tree θ contains no self-energy clusters. On the other hand D(x ), D(x1 ) ≤ 2−m+1 C0 , hence |x |, |x1 | ≤ 2−m+n+3 C0 by (4.4), and C0 |ν − ν 1 |−τ0 ≤ |x − x1 | ≤ 2−m+n+4 C0 , so that we get v∈V (T ) |ν v | ≥ (2−m+n+4 )−1/τ0 , which gives Nm (θ ) ≤ 1 + Em v∈V (θ) |ν v | − Em v∈V (T ) |ν v | − 1 ≤ Em v∈V (θ) |ν v | − 1, so that the bound is completely proved. Remark. The above discussion exploits the property that the tree θ that we consider cannot, by definition of renormalized tree, contain self-energy clusters, and follows Ref. [GG] which was based on the possibility of bounding the denominators proportionally to x 2 (in that case the proportionality factor was 1): a property also valid here for n ≤ n0 . For the bound on Nm (T ) we consider a subset G0 of the lines of a tree θ between two lines out and in . Set G = G0 ∪ in ∪ out . Let [pin ], [pout ] be the scales of the lines out and in , respectively, and suppose that pin , pout ≥ m, while all lines in G0 (if any) have scales [p] with p ≤ n − 1. Note that in general G0 is not even a cluster unless pin , pout ≥ n. Then we can prove that Nm (G0 ) ≤ max{Em v∈V (G0 ) |ν v | − 1, 0}, where V (G0 ) is the set of nodes preceding out and following in , and Em is defined above. If G0 has zero lines then the harmonic ν 0 of the (only) node in V (G0 ) is large, |ν 0 | ≥ 2(m−n−2)/τ0 (by the Diophantine property) and the statement is true. Hence we proceed inductively on the number of lines in G0 . If no line of G0 on the path P(G) connecting the external lines of G has scale [m] then the lines in G0 on scale [m] (if any) belong to trees with root on P(G), and the statement follows from the bound on trees. Suppose that ∈ P(G) is a line on scale [m], then call G1 and G2 the disjoint subsets of G such that G1 ∪ G2 ∪ = G. Then G1 ∪ and G2 ∪ have the same structure of G itself but each has less lines: and again the inductive assumption yields the result. R , the bound Therefore, as a particular case, by choosing G0 = T , with T ∈ Sk,n−1 for Nm (G) implies the bound on Nm (T ) we are looking for.
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The above analysis is taken from Ref. [Ge] and differs from Ref. [GG] because here the scales depend on ε and it is not clear how to define a “strong Diophantine condition”, which would allow a one-to-one correspondence between line scales and line momenta. R is then The bound on the contribution of a single self-energy cluster T ∈ Sk,n−1 m 1 0 ε 2 k k −2k 2k − 1 κ0 |ν v | (m+3)2Nm (T ) v 2 · (A3.3) 2 G C F e k! 0 0 m=0 1 ∞ 1 −m/τ ε 2 k Gk − 1 κ0 v∈V (T ) |ν v | 0E v∈V (T ) |ν v | ≤ · e− 2 κ0 v∈V (T ) |ν v | 22(m+3)2 , e 2 k! m=m0 +1
with F an upper bound on the constants F0 , F1 bounding the Fourier transform of the perturbation (see (1.4)), while m0 is defined so that log 2 m>m0 2(m + 3)2−m/τ0 E ≤ 21 κ0 and G0 , G are suitable constants. We have used that the number of lines with scale [−1] can be at most 21 k and their propagators are bounded below proportionally to (εa1 )−1 , so that we can treat separately the case m = −1 paying the price that εk has to be replaced 1 by Gk0 ε 2 k . The number of trees can be bounded by 4k k!, and the sum over the scale labels involves at most 2 possible values per line because of the upper and lower cut-offs present in the propagators definition. The sum over the harmonics can be estimated by making use of part of the exponential factor in (A3.3) (say 41 κ0 ) while the other 41 κ0 will be used as a 1
0 factor bounded by e− 4 κ0 2 , by (A3.1). Hence we get convergence at the exponential rate 2−1 for ε < ε1 (and ε1 is an explicitly computable constant) and the matrix M [n] (x; ε) is defined by a convergent series and it is bounded by (n−n−5)/τ
1
M [n] (x; ε) < Bε 2 e− 4 κ0 2
(n−n−4)/τ0
(A3.4)
,
for a suitable B which can be read from (A3.3), i.e. we get the first of the first line in (5.13) with the constant B replaced by B, τ = τ0 , and κ1 = 41 κ0 e−(n+4)/τ0 . The ε2 factor is due to the parallel remark that, in any self-energy cluster whose value contributes to M[n] (x; ε), k is certainly ≥ 2 (see the Remark to Definition 4 in Sect. 5). Therefore if ε is small enough (that is smaller than a constant independent of n ≤ n0 ), M[≤n] (x; ε) − M0 ≤ Bε 2
∞
1
e− 4 κ0 2
(n−n−4)/τ0
def
= B ε2 ,
(A3.5)
n=1
so that the eigenvalues of M[≤n] (x; ε) will be shifted with respect to the corresponding def
eigenvalues of M0 by γ ε 2 at most, with γ = B C, see (I) in Appendix A4. Hence if we define γ as B C and ε is chosen small enough, say ε < ε2 , so that γ ε2 < 21 εas 2−2n−2 (as it must be in order that the above argument be consistent, see the beginning of the current section) we obtain the validity of the assumed inductive hypothesis for all n ≤ n0 and of the first inequality in the first line of (5.13) where B can be chosen equal to B above. The symmetries in items (i) and (ii) are an algebraic consequence of the form of the Lindstedt series: hence they are a necessary consequence of the proved convergence, see Ref. [GG].
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A3.2. Smoothness. The function M [n] (x; ε) which we have just shown to be well defined for all ε small enough will be smooth in ε, x. We assume inductively that this is the case for M [p] (x; ε), 0 ≤ p < n − 1, and that the bounds in the first line of (5.13) hold for such p’s (the case p = 0 is obvious as M[0] (x; ε) ≡ M0 ). Each derivative with respect to x or, respectively, to ε will replace the value of a self-energy cluster with k nodes by a sum of k terms which can be bounded by a bound like (A3.3). In fact, given a self-energy cluster T , the right derivative ∂x+ may fall on a denominator of one of the k − 1 cluster lines. If its frequency is x + x0 with scale label [m], derivation yields, up to a sign, a product of two matrices ((x0 + x)2 − M[≤m] (x0 + x; ε))−1 times 2 (x0 + x) − ∂x± M[≤m] (x0 + x; ε) with an appropriate order of multiplication. The term 2 (x + x0 ) ((x0 + x)2 − M[≤m] (x0 + x; ε))−2 can be bounded proportionally to (C0−2 22(m−1) )3/2 ≤ (C0−2 22(m−1) )2 , while the remaining term can be studied by making −1/2 and it leads to the use of the inductive assumption ∂x M[≤m] (x0 + x; ε) ≤ Bε2 as −2 2(m−1) 2 ) , multiplied by Bε 2 .8 same bound found for the first term, i.e. (C0 2 If the derivative falls on either a ψp or a χp function, we can use that such derivative p m can be bounded proportionally to C0−1 2p and m−1 p=0 2 = 2 , to obtain again the same bound as the first case. Hence the final bound has the form B1 + ε 2 Bb with B1 , b suitable constants, provided ε is small enough, say ε < ε3 . The value of the constants B1 , b does not depend on the inductively assumed value for B: in particular B1 can be obtained (see Remark (2) below for a smarter bound) by replacing 2(m+3) in the two factors in the l.h.s. of (A3.3) by 22(m+3) and by inserting a factor k times a constant (to keep track of all the constant factors arising from differentiation). Therefore if B = 2B1 the estimate on ∂x+ M[≤n] (x; ε) follows if ε is small enough, say ε < ε4 . The same can be said about the left derivative ∂x− . The right and left differentiability of M[n] (x; ε) with respect to x is due to the dependence of M[n] (x; ε) on the function D(x): the latter has a discontinuous derivative at 9 a finite number of points (roughly at midpoints between the eigenvalues λ[0] j of M0 ). [n] Note that the denominators in the self-energy values defining M (x; ε) cannot vanish, and actually stay well away from 0, permitting the above bounds, because of the lower cut-off ψ0 (D(x)) appearing in the definition of the propagators g [0] (x; ε)(x; ε); see (5.6) and (5.7). The same argument holds for ∂ε : however the bound will be only Bε instead of Bε 2 because the derivative with respect to ε might decrease by one unit the degree of the self-energy values involved. Thus the first line of (5.13) is completely proved. Of course for each of the three terms we get a different constant B, but for simplicity we use for them all the largest, still calling it B. Remarks. (1) We could also prove existence of higher x, ε-derivatives of M[n] (x; ε) and of its eigenvalues λ[n] j (x, ε) for j > r via the above argument. (2) The more derivatives we try to estimate with the above method the smaller would become the set of allowed values of ε. This however is avoidable. Instead of imagining to include the bound C0−2 22m arising above as a consequence of the “extra” D(x + x0 ) or of the other derivatives into the factors 2m+3 associated with the divisors in (A3.3) Since the matrix M [m] (x0 + x; ε) is generated by self-energy clusters of degree at least 2. One could avoid having only left and right differentiability by using a regularized version of the function D(x) as discussed in Remark (2) after Lemma 2 in Sect. 5. 8 9
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0 one could simply further bound this by C0−2 22n and use part of the factor e− 4 κ0 2 (say replacing 41 κ0 with 18 κ0 ): this eventually leads to a bound on the s th right-derivative 1
(n−n−4)/τ
1
(n−n−4)/τ
0 with respect to x of a value of a self-energy cluster proportional to 2ns e− 8 κ0 2 but with an s-independent estimate of the radius of convergence (as the constant G in (A3.3) remains the same). This is sufficient to get the existence of the s th derivatives without any further restriction on ε: and a similar argument holds for the ε-derivatives.
A3.3. Cancellations. Only the bound in the fourth line of (5.13) follows from those in the first line. The bounds in the second and third lines express remarkable properties of Lindstedt series and are essentially algebraic properties: they are the “same” cancellations which occur in KAM theory, see Refs. [Ga2, GM1], and are based on the remark that if T is a self-energy cluster the entering and exiting lines have the same momentum ν: hence the sum of the harmonics of the nodes of T vanishes v∈V (T ) ν v = 0. We start by dealing with the trivial cases. Consider first self-energy clusters T such that |ν v | ≥ (C0 /26 |x|)1/τ0 . (A3.6) v∈V (T )
For such a self-energy cluster T one canuse part (say 1/8) of the exponential decay of κ0 −1/2τ1 the node factors to obtain a bound e− 8 v∈V (T ) |ν v | ≤ e−b1 |x| ≤ b2 x 2 , with b1 and 2 b2 two suitable positive constants, while a factor ε simply follows from the fact that any self-energy cluster has at least two nodes. So we can assume that (A3.6) does not hold. If ν is the momentum flowing in the entering line then the momentum flowing in a line ∈ (T ) of scale [p], p ≤ n, if the scale of the cluster is [n], will be ν 0 + σ ν, σ = 0, 1, where ν 0 is the momentum that would flow on if ν = 0. The corresponding frequency will be x = x0 + σ x, with obvious notations. Also self-energy clusters containing lines on scale [−1] along the path connecting the external lines can be dealt with in the same way. Indeed in such κ0 κ0 κ0 a case one has v∈V (T ) e 8 |ν v | ≤ e− 8 |ν| , with e− 8 |ν| ≤ b1 |ν|−2τ ≤ b2 x 2 , for suitable constants b1 and b2 . Also self-energy clusters containing either lines with momentum −ν or lines with momentum ν outside the path connecting the two external lines (such a situation is possible as more than one scale can be associated with each line) can be easily controlled by applying the same argument. The case in which there are lines with momentum ν along the path connecting the external lines can be discussed as follows. Let us consider the internal line of the self-energy cluster T with momentum ν which is the closest to the exiting line 1T . Then there must be at least one line between and 1T (that is preceding 1T and not preceding ) on the same scale as (otherwise there would be a self-energy cluster internal to T ), and of course ν = ν by construction. This means that v∈V (T ) |ν v | ≥ |ν − ν | is bounded from below proportionannly to 2n/2τ1 (by the κ0 second Diophantine conditions in (6.7)), hence v∈V (T ) e 8 |ν v | ≤ b1 C02 2−2n ≤ b2 x 2 , for suitable constants b1 and b2 . For all the other cases we shall need the cancellation mechanisms that we are going to describe. Consider first the case in which T does not contain any line on scale [−1] nor with momentum ±ν. Then, if the entering and exiting lines are imagined attached to the internal nodes of T in all possible ways (i.e. in k 2 ways if T contains k nodes) keeping all their labels unaltered then one obtains a family FT of self-energy clusters.
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If instead T contains at least one or more lines on scale [−1] and they are all outside the path between its external lines (so that we can not apply the argument given above) call T1 the connected subset of T containing no line on scale [−1] and containing the extrema of the external lines of T . Hence v∈V (T1 ) = 0: then we attach the entering and exiting lines to the nodes of T1 in all possible ways. Again we call FT the family so obtained. Note that each tree in such a family still has a line on scale [−1] along the path connecting its external lines. The contribution of each self-energy cluster of FT to each of the entries of the matrix M[n] (x; ε) with labels i, j ≤ r (the αα entries in the notations of Lemma 2) and with labels i ≤ r, j > r (the αβ entries) has the form Mi,j ;v,w (x, T ) νv,i νw,j or, respec tively, Mi,j ;v (x, T ) νv,i , with (1)
(2)
Mi,j ;v,w (x, T ) = Mi,j (T ) + xMi,j (T ) + x 2 Mi,j,v,w (x, T ), (1)
Mi,j ;v (x, T ) = Mi,j (T ) + xMi,j,v (x, T ),
i, j ≤ r, i ≤ r, r < j, (A3.7)
so that after performing the sum over the self-energy clusters of FT , i.e. after performing the sums v,w∈V (T ) or, respectively, v∈V (T ) (with T1 replacing T in the second case considered above), the first two terms in the first line and the first term in the second line do not contribute because v ν v = 0. However one has to show that the matrices M and M in the r.h.s. of (A3.7) satisfy appropriate bounds once the factors x determining the order of zero at x = 0 are extracted. From the convergence one expects that the bounds should still be proportional to ε2 while the derivatives ∂x± or ∂ε should satisfy bounds proportional to ε2 or to ε respectively. The (A3.7) are proved by means of interpolations, see Ref. [GM1], between the contributions of the self-energy clusters in the family FT . When we collect together the values of the self-energy clusters in FT then the arguments of some of the propagators can fall outside the supports of the respective cut-off function (because the lines are shifted but their scale labels are kept fixed so that scales of the propagators of the selfenergy clusters T ∈ FT are the ones inherited by T while the momentum flowing in them may change). This generates trees and clusters for which we made no estimates (because they are just 0). However when interpolating we may end up computing values of trees, with scale assignments which would give a value 0, at intermediate frequencies where the values no longer vanish. In estimating such interpolated values we can proceed as in the cases already treated, but it will not be necessarily true that a line of frequency x and scale [n] will satisfy 2−2(n+1) C02 < D(x). Nevertheless a slightly weaker version of this inequality has to hold in which the l.h.s. is divided by 4 (cf. also Remark (3) after the inductive assumption in Sect. 6), and the estimates will not only be possible but they can be regarded as already obtained because, as the reader can check, we have been careful in discussing the bounds obtained so far under such a weaker condition. This also clarifies why we have defined n in (4.2) one unit larger than what appeared there as necessary so that the estimate (4.4) is apparently worse than it should. In some cases, however, a serious problem seems to arise when actually attempting to derive bounds: namely the bounds on the matrices which appear as coefficients in (A3.7) can really be checked as just outlined by the above hints, and without affecting the values of ε for which one has convergence, only if x verifies the condition of being so small that the variations of the momenta flowing in the inner lines of T , when the entering or exiting lines are moved and re-attached to all nodes of T , remain so small that the quantities D(x ) corresponding to the lines in the cluster T stay essentially unchanged.
Degenerate Elliptic Resonances
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In certain cases shifting the entering or exiting lines to the nodes of the self-energy cluster T may considerably change the scales of the lines in T , but this is the case in which (A3.6) holds. And precisely in such a case the cancellations are not needed to prove the bound, because we have checked that the value of each self-energy cluster contributing to M [n] individually already verifies that bound that we want to prove. [p] If (A3.6) does not hold, then two cases are possible: either |x| is close to λj for some j > r or larger, and no cancellation occurs, or |x| is < C0 2−n . In the latter case the inequality opposite to (A3.6) implies that for ∈ (T ) one has |x0 | ≥ 4|x|, hence 2|x0 | ≥ |x | ≥ 21 |x0 |, so that the scales can change by at most one unit by shifting the external lines of T . Then the quantities D(x ) do not change much for all lines ∈ (T ), and we shall have the cancellation through the mentioned mechanism. Therefore the contribution of M[p] (x; ε) to M[≤n] (x; ε) can be bounded in both cases proportionally to 1 3 (n−n−4)/τ0 e− 4 κ0 2 times min{ε 2 , ε |x|2 } for the entries αα or times min{ε 2 , ε 2 |x|} for the αβ entries: either by the cancellation (second case) or by the general bound O(ε 2 ) on matrix elements (first case), because x 2 is of order O(ε). Finally we note that in the estimates of the M’s in (A3.7) we have to sum over the scale labels and this gives a factor per line larger than the one arising in the bound (A3.3) (which was 2); in fact we have to consider also trees with vanishing value: but the scales of the divisors associated with their lines can change at most by one unit with respect to the scale, hence we can have at most 4 scale labels per line. Remark. We stress once more that the above analysis holds if ε is small enough, say ε < ε 1 with ε 1 determined by collecting all the (three) restrictions imposed by requiring ε to be “small enough”, derived above and ε 1 is independent of n0 (otherwise it would be uninteresting). The reason is that as long as we do not deal with x’s which are too close to the eigenvalues of M0 , so that the key inequality (4.4) holds, we do not really see the difference between the hyperbolic and the elliptic cases: and in the hyperbolic cases there is no need for a lower cut-off at scale ∼ n0 where resonances between the proper frequencies (which are of order ε) and the elliptic normal frequencies become possible (as ε C02 2−2n0 ). A3.4. Resonant resummations. Concerning the proof of Lemma 6 we only need to add a few comments. The bounds on Nm (θ ) and Nm (T ) can be discussed exactly as for the scales [n] with n ≤ n0 , with the only difference that now one has to use also the second part of the Diophantine conditions (6.7), as already done in the argument leading to (6.12); in particular the role of the exponent τ0 is now played by 2τ1 (because of the Diophantine conditions in (6.7) which replaces (1.3) in the discussion), while in the analogues of (A3.1) and the following bounds no n appear, as the propagator divisors are bounded directly in terms of the corresponding scales, and not in terms of the frequencies. Also the argument given above about the cancellations extends easily to the scales [n], with n ≥ n0 . The only difference is that in (A3.6) the exponent 1/τ0 has
to be replaced −1] with 1/(2τ1 ), in such a way that for any line ∈ (T ) one has ||x0 | − |λ[n (ε)|| ≥ j 4|x|, hence the chain of inequalities
1 0 0 [n −1] [n −1] [n −1] (ε)| ≥ |x | − |λj (ε)| ≥ |x | − |λj (ε)| , 2 |x | − |λj 2 (A3.8)
follows, and again by shifting the external lines of T the scales of the internal lines can change at most by one unit, when (A3.6) is not satisfied
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Appendix A4. Matrix Properties (I) Let M0 be a d × d Hermitian matrix with eigenvalues λ1 < . . . < λp with multiplicities n1 , . . . , np and eigenspaces 1 , . . . , p on which we fix orthonormal bases ej,k , j = 1, . . . , p, k = 1, . . . , nj . Let M1 be Hermitian and consider the matrix M = M0 + εM1 . There exists a constant C such that, for ε small enough, there will be nj eigenvalues of M (not necessarily all distinct) which are analytic in ε and one has |λj,k (ε) − λj,k (ε)| ≤ Cε for k, k = 1, . . . , nj . zdz 1 Hint. If nj = 1 this follows immediately from the formula λj (ε) = Tr 2πi γj z−M , where γj is a circle around λj (0) of ε-independent radius smaller than half the minimum 1
separation δ between the λj for ε small enough (so that C1 ε d < δ for a suitable C1 )10 . Otherwise it follows from similar formulae for the projection operator Ej on j and for Ej MEj : dz z dz 1 1 , Ej MEj = Ej , Ej (A4.1) Ej = 2π i γj z − M 2πi γj z−M which, for ε small, can be expanded into a convergent power series in ε (as done explicitly in a similar context in (A4.3) below) because of the ε-independence of the radii of γj . One can also construct an orthonormal basis on j with vectors of the form q (q) vj,k = ej,k + ∞ q≥1 ε ej,k (simply applying the Hilbert-Schmidt orthonormalization to the vectors Ej ej,k , k = 1, . . . , nj ). One then remarks that the matrix Ej MEj has nj eigenvalues and that it has the form λj + ε M(ε). So the problem is reduced to the case in which M0 is the identity perturbed by an is proportional to the identity and there is nothing more to analytic matrix. Either M(ε) do, or it is not: hence there will be an order in ε at which the degeneracy is removed and repeating the argument we reduce the problem to a similar one for matrices of dimension lower than nj : and so on until we find a matrix (possibly one dimensional) proportional to the identity to all orders. In our analysis we need the following corollary. (II) Let M0 be Hermitian with r degenerate eigenvalues equal to 0 and s = d − r simple eigenvalues εaj , j = 1, . . . , s. (i) The matrix M0 + ε2 M1 with M1 Hermitian and differentiable in ε with bounded derivative will have s non-degenerate eigenvalues εaj + O(ε 2 ), j = 1, . . . , s, and r eigenvalues λ1 (ε), . . . , λr (ε), all analytic in ε, with the property that for all k = 1, . . . , r one has |λk (ε)| < C ε2 , if ε is small enough and C is a suitable constant. (ii) If M1 depends on a parameter x and is differentiable also in x with bounded derivative then |∂x λj (x; ε)| ≤ Cε2 ,
|∂ε λj (x; ε)| ≤ C, 1/r
|λj (x; ε) − λj (x ; ε)| ≤ Cε |x − x | 2
,
j > r, j ≤ r,
(A4.2)
if ε is small enough and C is a suitable constant. 10 Because the characteristic polynomials P (λ), P (λ) are related by P (λ) = P (λ) + εQ(λ, ε) with 0 0 Q of lower degree. Therefore there is L such that if |λ| > L then for all |ε| < 1 (say) it is P (λ) = 0. Furthermore if all roots of P differ by at least y from those of P0 one has |P (λ)| ≥ y d − εC d , where −1 C d = max||≤L,|ε|≤1 |Q(λ, ε). Hence y ≤ Cε d .
Degenerate Elliptic Resonances
357
The second relation in (A4.2) is not used in this paper and is given only for completeness. Hint. We apply the previous lemma to the matrices ε −1 M0 and εM1 and we get (i). To get (A4.2) we note that the x-derivative of M0 + ε 2 M1 is ε 2 ∂x M1 and the first of (A4.2) follows. To obtain the second of (A4.2) we have to compare the eigenvalues of M0 + ε2 M1 (x; ε) with those of M0 + ε2 M1 (x; ε) + ε2 O(|x − x |). By the above expression for the projection on the plane of the first r eigenvalues this is reduced to the problem of comparing two r × r matrices of order ε 2 and differing by O(|x − x |). The power 1/r arises from the estimate that the considered projection of the matrix M1 (which is only differentiable in x) has r eigenvalues close to 0 within C1 |x − x |1/r , for some C1 > 0 (by (I) above), and ε is small enough. Hence we get the second of (A4.2). A third property that we need is the following one. 2 2 ε x N ε2 xP (III) If M0 is as in (II) and M1 is Hermitian and has the form 2 ∗ 2 , with ε xP ε Q N, Q two r × r and r × s matrices and P a r × s matrix, then the first r eigenvalues of M0 + M1 are bounded by |λj (x, ε)| < Cε2 x 2 , for j = 1, . . . , r. Hint. This is obtained by using (A4.1) which gives the projection over the plane of the r eigenvalues within O(ε 2 ) of 0 as integral over a circle of radius 21 a1 ε, 1 E= 2πi
γ
k ∞ dz 1 M1 , z − M0 z − M0
(A4.3)
k=0
1 and one sees that (M1 z−M )k has for all k ≥ 1 the same form of M1 , with ε 2 replaced by 0 10 ε2k , so that the sum of the series is the matrix corresponding to the k = 0 term (it 00 is a d × d block matrix with the first r × r block 1 and the other blocks 0) plus a matrix of the same form of the basis vh = Eeh , h = 1, . . . , r consists of vectors 1 . Likewise M ε 2 x 2 uh , so that one checks that the matrix (vh , (M0 + M1 )vh ) is of the form eh + ε 2 xuh a r × r matrix which is proportional to ε 2 x 2 (i.e. it has the form ε 2 x 2 M2 (x, ε), with M2 bounded for ε small and for |x| < 1) and which, by construction, has the same eigenvalues as the first r eigenvalues of the matrix M0 + M1 .
For the above properties see also [RS, Ka]. Appendix A5. Algebraic Identities for the Renormalized Expansion We show that the function h defined through the renormalized expansion solves the equations of motion (1.5) for all ε ∈ E. This is essentially a repetition of Ref. [Ge]. We shall check that h = εg∂ϕ f (ψ + a, β 0 + b), where ϕ = (α, β) and g is the pseudo −2 iν·ψ h , differential ν ∞ operator with kernel g(ω · ν) = (ω · ν) . We can write h = ν∈Zr e hν = n=0 hn,ν (only two terms in this series are different from 0 for each ν), with ∞ R Val(θ ), where R hn,ν = k=1 θ∈ R k,ν (n) is the set of trees in k,ν such that k,ν (n) the root line has scale n. With respect to the previous sections we have dropped the component label γ ∈ {1, . . . , d} in the definition of the set of trees, for notational convenience.
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Note that, for all x = 0 and for all p ≥ 0 one has 1=
∞
ψn ([n] (x, ε))
n=p
n−1
χq ([q] (x, ε)),
(A5.1)
q=p
where the term with n = p has to be interpreted as ψp ([p] (x; ε)). The latter identity is checked as follows. One has 1 = ψp ([p] (x; ε)) + χp ([p] (x; ε)) (because ψ + χ ≡ 1); therefore 1 = ψp ([p] (x; ε)) + χp ([p] (x; ε))(ψp+1 ([p+1] (x; ε)) + χp+1 ([p+1] (x; ε))) (again because ψ + χ ≡ 1) and so on; since, fixed x, χn ([n] (x; ε)) ≡ 0 for n large enough (by Remark (8) to the inductive hypothesis, i.e. by (6.7)), then (A5.1) follows. [p] Set n (x; ε) = ψn ([n] (x; ε)) n−1 p=0 χp ( (x; ε)) for n ≥ 1, 0 (x; ε) = ψ0 ([0] (x; ε)): by using (A5.1) one can write, in Fourier space and evaluating the functions of ϕ at ϕ = (ψ + a, β 0 + b), ∞ g(ω · ν) ε∂ϕ f (ϕ) ν = g(ω · ν) n (ω · ν; ε) ε∂ϕ f (ϕ) ν n=0
= g(ω · ν)
∞
n (ω · ν; ε)(g [n] (ω · ν; ε))−1 g [n] (ω · ν; ε) ε∂ϕ f (ϕ) ν
n=0
= g(ω · ν) = g(ω · ν)
∞
(ω · ν)2 − M[≤n] (ω · ν; ε) g [n] (ω · ν; ε) ε∂ϕ f (ϕ) ν
n=0 ∞
∞ (ω · ν)2 − M[≤n] (ω · ν; ε)
n=0
Val(θ ), (A5.2)
k=1 θ∈ R (n) k,ν
R
where k,ν (n) differs from R k,ν (n) as it contains also trees which can have one renormalized self-energy cluster T with exiting line 0 , if 0 denotes the root line of θ ; for such trees the line entering T will be on a scale p ≥ 0, while the renormalized self-energy cluster T will have a scale nT = q, with q + 1 ≤ min{n, p}. The graphical representation in Fig. 5 makes the last step in (A5.2) clear: (ε∂ϕ f (ψ + a, β 0 + b)ν ), with h = (a, b) well defined and small by the analysis in Sect. 6, can be
Fig. 5. Here each of the lines exiting the bullets represents hν i , i = 1, . . . , p with h defined by the resummed series. Developing each h in a resummed tree series one realizes that the picture almost reconstructs h itself. However the trees obtained in this way may have internal lines of momentum ν , which together with the line 0 would form a self energy cluster. This is taken into account by extending the R domain of the summation from R k,ν (n) to k,ν (n)
Degenerate Elliptic Resonances
359
developed in Taylor series in h and then each h can be expressed as a tree sum with no self energy clusters which can be graphically represented as in the figure. Remark. Note that in both (A5.1) and (A5.2) only a finite number of addends in n is different from zero, as the analysis of Sect. 6 shows, so that the two series are well defined. The same observation applies to the following formulae, where appear series which, in fact, are finite sums. By explicitly separating in (A5.2) the trees containing such self-energy clusters from the others, ∞ ∞ g(ω · ν) ε∂ϕ f (ϕ) ν = g(ω · ν) (ω · ν)2 − M[≤n] (ω · ν; ε)
(ω · ν)2 − M[≤n] (ω · ν; ε) g [n] (ω · ν; ε)
∞
n=1 ∞ n−1
M [q] (ω · ν; ε)
∞
(A5.3)
Val(θ )
k=1 θ∈ R (p) k,ν
p=n q=0
+g(ω · ν)
Val(θ )
k=1 θ∈ R (n) k,ν
n=0
+g(ω · ν)
∞
(ω · ν)2 − M[≤n] (ω · ν; ε) g [n] (ω · ν; ε)
n=2 n−1 p−1
M [q] (ω · ν; ε)
∞
Val(θ ),
k=1 θ∈ R (p) k,ν
p=1 q=0
which, by the definitions of h, can be written as ∞ g(ω · ν) ε∂ϕ f (ϕ) ν = g(ω · ν) (ω · ν)2 − M[≤n] (ω · ν; ε) hn,ν
(A5.4)
n=0
+
+
∞
n (ω · ν; ε)
∞ n−1
n=1
p=n q=0
∞
n−1 p−1
n (ω · ν; ε)
n=2
M [q] (ω · ν; ε)hp,ν ! M
[q]
(ω · ν; ε)hp,ν .
p=1 q=0
The terms in the second line of (A5.4) can be written as p ∞ p−1 p=1
M [q] (ω · ν; ε)n (ω · ν; ε)+
q=0 n=q+1
=
∞ p−1 p=1 q=0
M [q] (ω · ν; ε)
p−1
∞
M [q] (ω · ν; ε)n (ω · ν) hp,ν
q=0 n=p+1 ∞ n=q+1
n (ω · ν; ε) hp,ν
(A5.5)
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and, by changing p → n and n → s, we obtain ∞ n−1 n=1
M [q] (ω · ν; ε)χ0 ([0] (ω · ν; ε)) . . . χq ([q] (ω · ν; ε)) ·
q=0
·
∞
χq+1 (
[q+1]
(ω · ν; ε)) . . . ψs ( (ω · ν; ε)) hn,ν (A5.6) [s]
s=q+1
=
∞ n−1
M [q] (ω · ν; ε)χ0 ([0] (ω · ν; ε)) . . . χq ([q] (ω · ν; ε)) hn,ν ,
n=1 q=0
where the identity (A5.1) has been used in the last line (with the correct interpretation of the term with s = j + 1 explained after (A5.1)). By the definition of the matrices M[≤n] (x; ε) one has n−1
M [q] (ω · ν; ε)χ0 ([0] (x; ε)) . . . χq ([q] (x; ε)) = M[≤n] (x; ε),
(A5.7)
q=0
so that, by inserting (A5.6) in (A5.3), after having used (A5.7), we obtain ∞ " (ω · ν)2 − M[≤n] (ω · ν; ε) g(ω · ν) ε∂ϕ f (ϕ) ν = g(ω · ν) n=0
+M
[≤n]
= g(ω · ν)
(ω · ν; ε) hn,ν
∞ n=0
(ω · ν)2 hn,ν =
∞
hn,ν = hν ,
n=0
so that the assertion is proved. Remark. Note that at each step only absolutely converging series have been dealt with, so that the above analysis is rigorous and not only formal. Acknowledgements. We are indebted to V. Mastropietro for many discussions and, in particular, to A. Giuliani for critical reading and several suggestions.
References [Ba] [Bo1] [Bo2] [Bo3] [Bo4] [BaG]
Barata, J. C. A.: On formal quasi-periodic solutions of the Schr¨odinger equation for a twolevel system with a Hamiltonian depending quasi-periodically on time. Rev. Math. Phys. 12(1), 25–64 (2000) Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Internatational Mathematics Research Notices 11, 475–497 (1994) Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995) Bourgain, J.: On Melnikov’s persistency problem. Math. Res. Lett. 4, 445–458 (1997) Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schr¨odinger equations. Ann. Math. 148(2), 363–439 (1998) Bartuccelli, M.V., Gentile, G.: Lindstedt series for perturbations of isochronous systems. A review of the general theory. Rev. Math. Phys. 14(2), 121–171 (2002)
Degenerate Elliptic Resonances
361
[BGGM] Bonetto, F., Gallavotti, G., Gentile, G., Mastropietro, V.: Lindstedt series, ultraviolet divergences and Moser’s theorem. Annali della Scuola Normale Superiore di Pisa Classe di Scienze 26(3), 545–593 (1998) [BKS] Bricmont, J., Kupiainen, A., Schenkel, A.: Renormalization group and the Melnikov problem for PDE’s. Commun. Math. Phys. 221(1), 101–140 (2001) [Ch1] Cheng, C.-Q.: Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems. Commun. Math. Phys. 177(3), 529–559 (1996) [Ch2] Cheng, C.-Q.: Lower-dimensional invariant tori in the regions of instability for nearly integrable Hamiltonian systems. Commun. Math. Phys. 203(2), 385–419 (1999) [ChW] Cheng, C.-Q., Wang, S.: The surviving of lower-dimensional tori from a resonant torus of Hamiltonian systems. J. Differ. Eqs. 155(2), 311–326 (1999) [CrW] Craig, W. , Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure and App. Math. 46(11), 1409–1501 (1993) [E1] Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Annali della Scuola Normale Superiore di Pisa Classe di Scienze 15(1), 115–147 (1988) [E2] Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions. Math. Phys. Electronic J. 2, paper 4, 1–33 (1996) (Preprint 1988) [F] Fefferman, C.: Pointwise convergence of Fourier series. Ann. Math. 98, 551–571 (1973) [Ga1] Gallavotti, G.: Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable hamiltonian systems. A review. Rev. Math. Phys. 6, 343–411 (1994) [Ga2] Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164(1), 145–156 (1994) [Ga3] Gallavotti, G.: Invariant tori: a field theoretic point of view on Eliasson’s work. In: Advances in Dynamical Systems and Quantum Physics, Ed. R. Figari, Singapore: World Scientific, 1995, pp. 117–132 [Ga4] Gallavotti, G.: Renormalization group in Statistical Mechanics and Mechanics: gauge symmetries and vanishing beta functions. Phys. Rep. 352, 251–272, (2001) [Ga5] Gallavotti, G.: Exact Renormalization Group. Paris IHP, 12 october 2002, Seminaire Poincar´e, Editors B. Duplantier, V. Rivasseau, Institut H. Poincar´e-CNRS-CEA [GG] Gallavotti, G., Gentile, G.: Hyperbolic low-dimensional tori and summations of divergent series. Commun. Math. Phys. 227(3), 421–460 (2002) [GBG] Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of the ergodic, qualitative and statistical properties of motion. Berlin: Springer–Verlag, 2004 [Ge] Gentile, G.: Quasi-periodic solutions for two-level systems. Commun. Math. Phys. 242(1–2), 221–250 (2003) [GM1] Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8(3), 393–444 (1996) [GM2] Gentile, G., Mastropietro, V.: Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method. J. de Math´ematiques Pures et Appliqu´ees 83(8), 1019–1065 (2004) [GMP] Gentile, G., Mastropietro, V., Procesi, M.: Periodic solutions for completely resonant wave equations. Commun. Math. Phys., to appear, DOI: 10.1007/s00220-044-1255-8 [JLZ] Jorba, A., de la Llave, R., Zou, M.: Lindstedt series for lower-dimensional tori. In: Hamiltonian systems with three or more degrees of freedom (S’Agar´o, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Ed. C. Sim´o, Dordrecht: Kluwer Acad. Publ., 1999, pp. 151–167 [Ka] Kato, T.: Perturbation theory for linear operators. Grundlehren der Mathematischen Wissenschaften, Band 132, Berlin-New York: Springer-Verlag, 1976 [Ku1] Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Akademiya Nauk SSSR. Funktsional ny˘ı Analiz i ego Prilozheniya 21(3), 22–37 (1987) [Ku2] Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics 1556, Berlin: Springer-Verlag, 1993 [KP] Kuksin, S.B., P¨oschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schr¨odinger equation. Ann. Math. 143(1), 149–179 (1996) [LW] de la Llave, R., Wayne, C.E.: Whiskered and low dimensional tori in nearli integrable Hamiltonian systems. Math. Phys. Electronic J. 2004 [Me1] Mel’nikov, V.K.: On certain cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function. Doklady Akademii Nauk SSSR 165, 1245–1248 (1965); English translation in Sov. Math. Doklady 6 , 1592–1596 (1965) [Me2] Mel’nikov, V.K.: A certain family of conditionally periodic solutions of a Hamiltonian systems. Doklady Akademii Nauk SSSR 181, 546–549 (1968); English translation in Sov. Math. Doklady 9, 882–886 (1968)
362 [Mo] [P1] [P2] [RS] [R] [T] [WC] [Wa] [XY] [Y]
G. Gentile, G. Gallavotti Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Annalen 169, 136–176 (1967) P¨oschel, J.: On elliptic lower-dimensional tori in Hamiltonian systems. Math. Zeits. 202(4), 559–608 (1989) P¨oschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comm. Math. Helv. 71(2), 269–296 (1996) Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York-London, Academic Press, 1978 R¨ussmann, H.: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Reg. and Chaotic Dyn. 6, 119–204 (2001) Treshch¨ev, D.V.: A mechanism for the destruction of resonance tori in Hamiltonian systems. Rossi˘ıskaya Akademiya Nauk. Matematicheski˘ıSbornik 180(10), 1325–1346 (1989); English translation in Math. of the USSR-Sbornik 68(1), 181–203 (1991) Wang, S., Cheng, C.-Q.: Lower-dimensional tori for generic Hamiltonian systems. Chinese Sci. Bull. 44(13), 1187–1191 (1999) Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990) Xu, J.,You, J.: Persistence of lower dimensional tori under the first Melnikov’s non-resonance condition. J. de Math. Pures et Appl. 80(10), 1045–1067 (2001) J. You: Perturbations of lower-dimensional tori for Hamiltonian systems. J. Diff. Eqs. 152(1), 1–29 (1999)
Communicated by A. Kupiainen
Commun. Math. Phys. 257, 363–394 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1294-9
Communications in
Mathematical Physics
String Scattering from Decaying Branes Vijay Balasubramanian1 , Esko Keski-Vakkuri2 , Per Kraus3 , Asad Naqvi4 1
David Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, PA 19104, USA. E-mail: [email protected] 2 Helsinki Institute of Physics and Department of Physical Sciences, University of Helsinki, P. O. Box 64, CA, 00014, Finland. E-mail: [email protected] 3 Department of Physics, University of California, Los Angeles, CA 90095, USA. E-mail: [email protected] 4 Institute for Theoretical Physics, University of Amsterdam, The Netherlands. E-mail: [email protected] Received: 24 May 2004 / Accepted: 15 July 2004 Published online: 25 February 2005 – © Springer-Verlag 2005
Abstract: We develop the general formalism of string scattering from decaying Dbranes in bosonic string theory. In worldsheet perturbation theory, amplitudes can be written as a sum of correlators in a grand canonical ensemble of unitary random matrix models, with time setting the fugacity. An approach employed in the past for computing amplitudes in this theory involves an unjustified analytic continuation from special integer momenta. We give an alternative formulation which is well-defined for general momenta. We study the emission of closed strings from a decaying D-brane with initial conditions perturbed by the addition of an open string vertex operator. Using an integral formula due to Selberg, the relevant amplitude is expressed in closed form in terms of zeta functions. Perturbing the initial state can suppress or enhance the emission of high energy closed strings for extended branes, but enhances it for D0-branes. The closed string two point function is expressed as a sum of Toeplitz determinants of certain hypergeometric functions. A large N limit theorem due to Szeg¨o, and its extension due to Borodin and Okounkov, permits us to compute approximate results showing that previous naive analytic continuations amount to the large N approximation of the full result. We also give a free fermion formulation of scattering from decaying D-branes and describe the relation to a grand canonical ensemble for a 2d Coulomb gas.
1. Introduction D-branes, as solitons of open string theory that are localized in space, have given many insights into nonperturbative phenomena in string theory such as string duality, and resolve many timelike singularities of General Relativity including those inside some black holes. In addition, they lead to the holographic description of 11 dimensional asymptotically flat space via a Matrix model and of spaces with a negative cosmological constant in terms of a dual conformal field theory. Many of these developments, and particularly the last, arose from an understanding of D-brane dynamics – namely how
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closed strings scatter from such solitons which are quantized in terms of open string fluctuations. It is of great interest to understand similar issues in the time dependent context of rapidly expanding universes, particularly in view of the likely occurrence of a Big Bang followed by inflation in the early universe and the possible presence of a positive cosmological constant > 0 now. Exploration of the symmetries and structure of universes with positive has suggested that if they have a holographic description the dual might be related to a Euclidean CFT living on the early or late time boundaries of such spacetimes ([1, 2]).1 Time, in such a picture, emerges holographically via the RG flow of the Euclidean field theory dual in analogy with the emergence of the radial direction of AdS space from the RG flow of a Lorentzian field theory ([1, 2]). In order for such a picture to be actually realized in string theory one would need some kind of D-brane localized in time, called an S-brane in [3], with a decoupling limit relating the closed strings on the spacetime background to the open strings quantizing the brane. One might hope that the spacetime near such a Euclidean brane would be rapidly expanding by analogy with the rapid expansion of the transverse space in the vicinity of a conventional brane. In the decoupling limit for standard D-branes it is precisely this rapid transverse expansion that gives rise to the exponential increase in the volume of AdS spaces as the boundary is approached. Thus one might hope that a Euclidean brane would lead to a spacetime exponentially expanding in time in a suitable decoupling limit. Sen has proposed [4] that S-branes are concretely realized in string theory by the exact boundary CFTs representing decaying branes. In bosonic string theory the exactly marginal boundary interaction 0 Sbndy = λ dt eX (1.1) (where X0 is the timelike scalar) describes such a brane [5–7].2 The basic results that have been obtained so far are: 1. At vanishing gs the brane decays to “tachyon matter” with energy but vanishing pressure [4]. 2. At finite coupling there is tremendous production of very heavy non-relativistic closed strings [8]. 3. It has been proposed that the resulting coherent state of heavy closed strings is an equivalent description of the tachyon matter. 4. In view of (2) and (3) it is proposed that there is a new open-closed duality hinting at a new kind of holography [9, 10]. 5. There is a close relation with the dynamics of decaying unstable branes in the c = 1 matrix model that suggests that the picture (1)–(4) is essentially correct ([11–14]). To understand the structure of tachyon matter, and to explore the possibility of a decoupling limit leading to timelike holography, a central problem is to compute general scattering amplitudes of closed and open strings from the decaying brane. Closed strings can be used particularly to probe the structure of the tachyon matter final state that exists 1
One heuristic way of motivating this is to note that Lorentzian de Sitter space and Euclidean AdS space are solutions to −X02 + X12 + · · · Xd2 = ±l 2 embedded in flat (d + 1) dimensional space. Both hyperboloids are thus different real sections of the same complex manifold. 2 There are also cosh X 0 and sinh X 0 interactions, which describe brane formation and subsequent decay [4]. Note the analogy with the inflationary and global parametrizations of de Sitter space, namely ds 2 = −dt 2 + et d x2 versus ds 2 = −dt 2 + cosh(t)2 d2 .
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after the brane decays, and open strings, since they will only be present in the spectrum at early times, can be used to change the initial brane configuration which is decaying. Thus, in Sect. 2 and 3 we develop the general formalism of string scattering from decaying branes in bosonic string theory. In a perturbative approach, amplitudes can be written as a sum of correlators in a grand canonical ensemble of unitary matrix models, with time setting the fugacity. If we integrate over the zero-mode time coordinate at the outset, the amplitudes are connected by analytic continuation to a single unitary matrix model, the rank of which is related to the total energy carried by the external vertex operators. Based on the approach which was employed with success in bulk Liouville theory [15], it was proposed (in [7, 16]) to compute bulk correlation functions by “analytically continuing” the vertex operator momenta from discrete integer values where direct computation is simple. This procedure has long been known to be somewhat questionable since an analytic function cannot be defined from just the data at a discrete set of points without additional constraints such as knowledge of the behavior at infinity or consistency conditions imposed by symmetries. In the case of Liouville theory, suitable consistency conditions were found (see the review [17]) and one can try to exploit the resulting techniques to explore the physics of decaying branes ([7, 18–21]). For full branes, an interesting prescription to compute n-point closed string disk amplitudes was given in terms of viewing the brane as an array of ordinary D-branes in imaginary time [22] (for generalization to higher genus, see [23]). In this paper, we describe an alternative formulation of the scattering amplitude calculation which is well-defined in an open subset of the complex energy plane and therefore permits reliable analytic continuation. We will study in detail the open-closed and closed-closed two point functions, and compare our results to previous work. In Sect. 4, we study the emission of closed strings from a decaying brane with initial conditions perturbed by the addition of an open string vertex operator.3 Using an integral formula due to Selberg, the relevant amplitude is expressed in closed form in terms of zeta functions. Perturbing the initial state in this way can either suppress or enhance the emission of high energy closed strings for extended branes, but always enhances it for D0-branes. This is consistent with the picture that D0-branes decay entirely into closed strings, with very heavy closed strings making up the tachyon matter state, while higher dimensional branes can decay partly into unstable lower dimensional branes. However, the enhancement of emission that we find in some cases will increase the direct closed string emission even by the higher dimensional branes. We discuss the consequences for the hypothesis that tachyon matter is nothing but a state of very heavy closed strings. In Sect. 5, the closed string two point function is expressed as a sum on N of Toeplitz determinants of certain N × N matrices of hypergeometric functions. A large-N limit theorem due to Szeg¨o, and its extension due to Borodin and Okounkov, permit us to compute approximate results that show that previously used methods to compute scattering amplitudes in the decaying brane amount to a leading large N approximation from our perspective. In Appendix C we give a free fermion formulation of scattering from decaying branes by extending old techniques of Douglas [24]. In Appendix D we describe the relation of the decaying brane correlators to a grand canonical ensemble for a classical Coulomb gas.
3
Another approach to these amplitudes will be described in [21].
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V. Balasubramanian, E. Keski-Vakkuri, P. Kraus, A. Naqvi
2. Review of Closed String Tachyon Scattering from D-Branes To establish notation and conventions, in this section we review the standard computation of closed string tachyon scattering from a D-brane. This amounts to computing the bulk two point function on the disk, with Neumann and Dirichlet boundary conditions in the various directions. 2.1. Kinematics. We follow the conventions of Polchinski [25], with ηµν = (−, +, +, + , . . . ), and α = 1. The closed string tachyon vertex operator is eik·X ,
k 2 = −m2 = 4.
(2.1)
For scattering from a Dp-brane, we divide the momenta into the p + 1 parallel components k , and the 25 − p transverse components k ⊥ . We write the momenta of the two closed string tachyons as k1 = (k , k1⊥ ) and k2 = (−k , k2⊥ ). The Mandelstam variables are defined as s = 2(k )2 ,
t = k 1 · k2 .
(2.2)
We can factorize the amplitude in the closed or open string channels, and find poles at the location of physical string states: closed: k 2 = −m2 = −4(N − 1), open: k 2 = −m2 = −(N − 1)
(2.3)
with N = 0, 1, 2, . . . . This implies that poles occur at closed: t = −2(N + 1), open : s = −2(N − 1).
(2.4)
2.2. Correlators on disk. We work on the unit disk, |z| ≤ 1. As is standard, we separate out the zero modes from the worldsheet fields X µ (z, z¯ ) by writing
Xµ (z, z¯ ) = x µ + X µ (z, z¯ )
(2.5)
with d 2 z X µ (z, z¯ ) = 0. The zero mode integrals are done at the end of the calculation, and enforce momentum conservation in Neumann directions. The Neumann and Dirichlet correlators on the disk are4 − 21 ηµν ln |z − w|2 + ln |1 − zw| ¯ 2 N µ ν X (z, z¯ )X (w, w) ¯ = (2.6) − 21 ηµν ln |z − w|2 − ln |1 − zw| ¯ 2 D. 4 In fact, there is a subtlety in defining the Neumann correlator. The Green’s function which obeys Neumann boundary conditions and vanishes when integrated over the disk is − 21 ηµν ln |z − w|2 + ln |1 − zw| ¯ 2 − z¯z − ww¯ − c , where c is a number which depends on the choice of worldsheet metric. The extra terms drop out after using spacetime momentum conservation (see Sect. 6.2 of [25]), and so don’t contribute to scattering from an ordinary D-brane. However, such terms do seem to contribute when considering D-branes with nontrivial worldvolume fields, as is our interest. Nevertheless, following standard practice, we will continue to drop these terms by adopting this as our prescription for defining the (naively divergent) X0 path integral.
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For z in the bulk and w on the boundary (w = eit ): µν µ ν −η ln |z − eit |2 N X (z, z¯ )X (t) = 0 D.
(2.7)
For both points on the boundary:
X µ (t1 )X ν (t2 ) =
−ηµν ln |eit1 − eit2 |2 N 0 D.
For self-contractions in the bulk: µ
ν
X (z, z¯ )X (z, z¯ ) =
(2.8)
− 21 ηµν ln |1 − z¯z|2 N 1 µν ln |1 − z¯z|2 D. 2η
(2.9)
For self-contractions on the boundary: µ
ν
X (t)X (t) =
0 N 0 D.
(2.10)
2.3. Closed string tachyon scattering from static D-brane. Using the above correlators, the two point function of bulk exponentials is
¯ eik1 ·X (z,¯z) eik2 ·X (w,w) = |z − w|t |1 − zw| ¯ −s−t |1 − z¯z| 2 −2 |1 − w w| ¯ 2 −2 . (2.11) s
s
The S-matrix amplitude is given by fixing one vertex operator at the origin, z = 0, and integrating w over the disk. Up to an overall constant and a momentum conserving delta function, the amplitude is 1 2t + 1 2s − 1 t 2 2s −2 , (2.12) dr r r (1 − r ) = 2s + 2t 0 which indeed exhibits poles in accord with (2.4). In the remainder of this work we are interested in computing the scattering amplitude from a decaying D-brane. Physically, as a function of x 0 , we expect to find an amplitude which interpolates between that of a D-brane and that of a collection of closed strings (or tachyon matter) into which the D-brane decays. One signature of this is that the open string poles should be absent at late times. Besides this, it is difficult to exhibit any “smoking gun” signatures of the presence/absence of the D-brane. For instance, it is known that in the high energy and fixed angle regime, both types of amplitudes behave 2 as e−E f (θ) , indicating softness at short distance. 3. Scattering from the Rolling Tachyon 0
Now we consider the tachyon profile T (X0 ) = λeX corresponding to the boundary interaction e−Sbndy = e−λe
x0
dt eX
0
,
(3.1)
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V. Balasubramanian, E. Keski-Vakkuri, P. Kraus, A. Naqvi
where we have separated out the zero mode as in (2.5). Consider the scattering of bulk tachyons. The X 0 part of the computation involves A =
DX 0 e−S
eiωa X
0 (z ,¯ a za )
a=1
=
dx 0 eix
0
a
ωa
e−λe
x0
dt eX
0
eiωa X
0
(za ,¯za )
.
(3.2)
a=1
The full amplitude also contains terms involving the spacelike Xi that are the same as for a standard D-brane, as well as integrals over the locations of the vertex operators za . As we will describe below the bulk vertex operators can be moved to the boundary (|z| = 1), to compute amplitudes with arbitrary numbers of open string tachyon vertex operators. We can always choose a gauge in which our vertex operators (assuming they carry nonzero energy) contain no timelike oscillators [26]. Thus the interesting part of any correlation function (e.g., for closed string scattering from a brane with a perturbed initial state) involves interactions with the boundary tachyon that are summarized by (3.2).
3.1. Perturbative approach and matrix integral formulation. One approach is to expand (3.2) in powers of the boundary interaction; i.e. as a power series in λ. The magnitude of λ can be changed by shifting x 0 , so truncating the power series is only sensible when considering scattering processes which are dominated at early times. In general, we must keep and sum the entire series. For the cases we consider, the sum will have a finite radius of convergence. Expanding, we obtain A =
0 ix 0
dx e
a
ωa
0 ∞ (−2πλex )N N!
N=0
N 0 dti X 0 (t1 ) X 0 (tN ) × ...e eiωa X (za ,¯za ) . e 2π i=1
(3.3)
a=1
By separating out the x 0 integral in this way we are isolating the contribution to the total scattering amplitude from the partially decayed state of brane that is present at any particular time. The late time contribution from x 0 → ∞ should isolate the effects of the tachyon matter final state to which the brane is supposed to decay. To calculate the fixed x 0 contributions we now need to evaluate (N) a
N 0 0 dti X 0 (t1 ) e = . . . eX (tN ) eiωa X (za ,¯za ) . 2π i=1
(3.4)
a=1
The Wick contractions are straightforwardly evaluated using the Green’s functions in (2.6)–(2.9), yielding
String Scattering from Decaying Branes
(N) a
=
369
−ωa ωb
|za − zb |
a
− 21 ωa ωb
|1 − za z¯ b |
ab
N dti iti itj 2 −iti 2iωa . × |e − e | |1 − za e | 2π i=1
i<j
(3.5)
ia
An elegant way of rewriting (3.5) is in terms of matrix integrals [6, 16]. In particular, dti iti itj 2 the measure N i<j |e − e | is nothing but the measure for U (N ) matrices i=1 2π expressed in the eigenvalue basis, the product of exponentials being the Vandermonde determinant. This leads to the identification
(N) −ωa ωb − 21 ωa ωb a = N! |za − zb | |1 − za z¯ b |
a
×
dU U (N)
ab
|det(1 − za U )|2iωa ,
(3.6)
a
where U is a unitary N ×N matrix, and we have normalized the measure to We can now write
1 A = |za − zb |−ωa ωb |1 − za z¯ b |− 2 ωa ωb a
×
U (N) dU
= 1.
ab 0 ix 0
dx e
a ωa
F (z1 , ω1 , . . . , z , ω ; µ),
(3.7)
where we have isolated the relevant summation by defining F (z1 , ω1 , . . . , z , ω ; µ) =
∞ N=0
e−Nµ
dU U (N)
| det(1 − za U )|2iωa
(3.8)
a
and µ = −x 0 − ln(−2π λ).
(3.9)
Here µ plays the role of a complex chemical potential, controlling the weights of the various U (N) matrix integrals. As we will discuss in Appendix D there is indeed a precise correspondence to the grand canonical ensemble of a statistical mechanical system. Note that the late time behavior corresponds to Re(µ) → −∞. The essential problem is to compute the matrix integrals appearing in (3.8), perform the sum over N , and evaluate the x 0 integral. Finally, to compute a string theory S-matrix element, we must also integrate over the positions of the vertex operators. Having obtained (3.7) for general bulk tachyon amplitudes, it is simple to include boundary tachyons. Simply take any number of vertex operators to the boundary, za → 1, 1 2 and remove the corresponding factors of |1 − za z¯ a |− 2 ωa , since there are no boundary self-contractions according to (2.10).
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3.2. Integrated approach: performing the zero mode integral first. An alternative approach is based on a strategy which was employed with success in bulk Liouville theory [15]. Instead of leaving the x 0 integration until the end, we can perform it at the outset. This approach requires analytic continuation in the momenta and can miss non-analytic pieces that are known to be present in the amplitude. Nevertheless, useful information can be obtained. So let us return to (3.2) and perform the x 0 integral using ∞ x0 0 dx 0 e−αe eiωx = α −iω (iω) (3.10) −∞
to obtain Aint
= (2π λ)
−i
a ωa
(i
a
−i dt 0 eX (t) ωa ) 2π
a
ωa
eiωa X
0
(za ,¯za )
. (3.11)
a=1
Using the identity (1 − iz)(iz) = −iπ/ sinh πz we can write this as
(2πλ)−i a ωa
B , = −iπ sinh(π a ωa ) −i a ωa dt 0 0 1
B = eiωa X (za ,¯za ) . eX (t) (1 − i a ωa ) 2π
Aint
(3.12)
a=1
As we will see, the overall factor in Aint is of exactly the form expected for the closed string one-point function [8]. The expression for B contains complex powers of the fields and therefore must be defined by appropriate analytic continuation. In particular,
to apply standard techniques, we need −i a ωa to be a positive integer. We will now follow the procedure of defining B by continuing the external momenta to imaginary integer values and then “continuing” back. This procedure requires some care, since in general the values of a function at a discrete set of arguments does not determine its behavior in the entire complex plane even with an assumption of analyticity. However, we will find situations in which the expression for B , evaluated formally for imaginary integer momenta, is actually defined on some open subset of the complex ω plane, thus permitting reliable analytic continuation. Proceeding in this fashion, we perform the continuation iωa → −na , to non-negative integers na , finding N 1 dtj X 0 (tj ) −na X 0 (za ,¯za ) e e B = N! 2π j =1
(3.13)
(3.14)
a=1
with N = a na . The correlator is the same as in (3.4) after the substitution (3.13). The result is therefore
1 |za − zb | 2 na nb |1 − za z¯ b |na nb B = a
×
dU U (N)
a
ab
|det(1 − za U )|−2na .
(3.15)
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Comparing with (3.7) an essential difference is that there is no longer a sum over N which needs to be evaluated, and the exponents in (3.15) are restricted to special integer values (which we’ll eventually have to continue to generic values). At these imaginary integer momenta, and for sufficiently large N , the matrix integral (3.15) is N independent, and can be efficiently evaluated group
theoretically [16] or using SU(2) current
algebra [7]. As shown in [16], for N ≥ a na (which applies to our case, since N = a na ), | det(1 − za U )|−2na = |1 − za z¯ b |−na nb . (3.16) dU a
ab
This then yields B =
|za − zb |na nb
a
1
|1 − za z¯ b |− 2 na nb .
(3.17)
ab
The final step is to “undo” the analytic continuation, by replacing na = −iωa in (3.17), which, as we have stressed, cannot be justified without additional information. Nevertheless, if we optimistically apply this procedure we are led to:
Aint
1 (2πλ)−i a ωa
= −iπ |za − zb |−ωa ωb |1 − za z¯ b | 2 ωa ωb . sinh(π a ωa ) a
(3.18)
ab
In Sect. 4 we will see that naively using (3.16) and then analytically continuing to real momenta leads to unphysical results for bulk-boundary amplitudes and we will describe a better procedure for computing B . In Sect. 5 we discuss different procedures for computing bulk-bulk amplitudes.
3.3. Comparison of one-point functions. It is helpful to compare the results for the tachyon one-point function ( = 1) obtained in the two approaches.5 By a conformal transformation we can take the bulk vertex operator to be at the origin of the disk, z = 0. 0 In the perturbative approach we find from (3.8) that F = 1/(1 + 2π λex ), and therefore A1 =
dx 0
eiωx
0
1 + 2πλe
x0
= −iπ
(2π λ)−iω . sinh π ω
(3.19)
Comparing with (3.18) we then find an agreement between the two approaches: A1 = Aint 1 . Note that in the perturbative approach the sum over N is convergent only for 0 |2πλex | < 1, but we define it in general by analytic continuation. The analytically continued sum vanishes exponentially for large x 0 , which yields a convergent x 0 integral. The finite radius of convergence of the sum over N is a feature of all the amplitudes we have examined, and makes it challenging to extract the late time behavior. In particular, to find the late time behavior of the unintegrated amplitude we must first find the exact early time behavior, so that we have an exact formula to analytically continue. This in turn means that we must perform an exact summation of the perturbation series. 5 This computation also applies to an arbitrary closed string state, since as pointed out in [8], we can always choose a gauge such that there are no timelike oscillators.
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In the calculation above this turned out to be easy, but the requisite sums become more challenging for higher point amplitudes. In light of this, it would be extremely useful to develop a method to access the late time behavior directly without resorting to analytic continuation. Although we have so far focused on tachyon vertex operators, it is instructive to consider a couple of other examples. Using the perturbative approach we can compute 0 the one-point function of ∂X 0 ∂X 0 eiωX at the origin to be [6]
0 eiωx −Sbndy 0 0 iωX0 0 1 iωx 0 e ∂X ∂X e = dx −e 2 1 + 2π λex 0 =−
iπ (2πλ)−iω − 2π δ(ω). 2 sinh πω
(3.20)
The δ(ω) terms come from the N = 0 term in the sum, which is special since the bulk fields have nothing to contract against. On the other hand, if we repeat this calculation in the approach where we first integrate over x 0 we will miss this δ(ω) term. This is because the latter approach is based on analytic continuation in ω and can therefore miss non-analytic pieces. The δ(ω) term can be shown to be responsible for energy conservation: it gives T00 where Tµν is the energy-momentum tensor of the decaying brane. The lesson is that the integrated approach can miss certain physically relevant non-analytic contributions, but can still be useful, provided we keep this lesson in mind. Further subtleties are revealed when we study the one-point functions of higher dimension operators corresponding to massive string modes. A finite number of terms in the sum over N will be “special”, leading to integrands which diverge for late times 0 as epx for positive integer p (this behavior was observed in [27, 28].) Such terms will also arise in amplitudes involving more than one bulk tachyon, since such operators will appear in the OPE. This behavior clearly makes the direct evaluation of x 0 integrals problematic. To define these integrals one should impose the constraints of conformal invariance, as explained in [29, 30]. In particular, after computing the tachyon one-point function, we can write the remaining operators as Virasoro descendants, and then use the fact that boundary state is conformally invariant to generate the remaining amplitudes. 4. Bulk-Boundary Two-Point Function: Perturbing the Initial Brane State A two-point amplitude in which one of the bulk operators is moved to the boundary describes an open-closed scattering amplitude. The closed string could either be an instate or out-state, but because the D-brane is decaying and does not exist at late times, the open string vertex operator can only be describing an in-state. Thus such amplitudes provide a systematic way of exploring the effects of perturbations to the initial conditions describing a decaying brane. For example, we could construct a localized lump on the brane at early times and ask how long in time the effect of this deformation lasts. In the previous sections, we have focused on closed string tachyons. However, the generalization to include an arbitrary closed string state is relatively straightforward if we choose a gauge in which the on-shell closed string state with non-zero energy has no time-like oscillators [26]. In this gauge, the closed string vertex operator has the form 0
V = eiωc X Vsp .
(4.1)
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The spatial part of this operator Vsp is constructed from the 25 spatial fields and is a ω2
Virasoro primary with conformal dimension = 1 + 4c . The computation of a bulkboundary amplitude factorizes into a product of the two point function in the time-like boundary Liouville theory and in the free spatial CFT. The form of the closed string vertex operator (4.1) is useful, because in the time-like boundary Liouville CFT the vertex operator has the same form as the tachyon vertex operator. In the spatial direction, the operator is non-trivial. We can write Vsp as X · · · , ∂¯ X, ∂¯ 2 X · · · )ei p· ∂ 2X , Vsp = P(∂ X,
(4.2)
for a polynomial P. In computing the bulk-boundary two point function with such an arbitrary closed string state, the contribution from the spatial part of the CFT is given by X(0) · · · , ∂¯ X, ∂¯ 2 X · · · )ei p· ∂ 2X : : ei k X(t) : , : P(∂ X,
(4.3)
where t parameterizes the position of the open string vertex operator on the boundary. The integration over the zero modes in the Neumann directions will yield a factor of δ(p + k ). This δ-function will be multiplied by terms from the contractions of the bulk and boundary fields. For no open string vertex operator (or equivalently for k = 0) the result of these contractions is just a phase [8]. In particular, note then that the absolute square of the result is independent of ωc , even though ωc enters into the definition of ω2
Vsp via = 1 + 4c , a result which is most easily seen from the boundary state. In the case with the open string vertex operator there will be additional contributions from the various ways of contracting fields from the bulk and boundary operators. This result and therefore on ωo . However, will in general depend on the open string momentum k, the squared result will again be independent of ωc just as before: ωc dependence can potentially enter only in the Dirichlet part of the computatation, but since the open string vertex operator has no momentum components in the Dirichlet directions, this part of the computation is identical to that without the open string vertex operator [8]. So as far as the spatial CFT is concerned, the effect of the open string vertex operator is to contribute a ω0 and P dependent factor, but not to alter the ωc dependence. Since our primary interest is in the ωc dependence, we will omit the extra ω0 and P dependence (though these could be straightforwardly computed), and so the formulas we will write below are strictly valid only for P = 1, i.e. for the closed string tachyon. We now return to the computation of the bulk-boundary two point function in the timelike boundary Liouville theory. First consider the perturbative approach. To obtain a correlator with one bulk and one boundary operator from (3.7), we set = 2, use conformal invariance to place one operator at z1 = 0 and the other at z2 = 1,6 and discard the diver1 2 gent factor (1 − z2 z¯ 2 )− 2 ω2 since it is removed by boundary normal ordering. This gives 0 (4.4) A2 (ωc , ωo ) = dx 0 ei(ωc +ωo )x F (0, ωc , 1, ωo ; µ), with F (0, ωc , 1, ωo ; µ) =
∞
e−Nµ IN (ωo ),
(4.5)
N=0 6 Actually, this step is not entirely innocent. Separating out the zero mode x 0 breaks conformal invariance, and so conformal invariance can only be restored after performing the x 0 integration. Since x 0 is noncompact, there can be subtleties associated with boundary terms, as discussed in [31].
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and
IN (ωo ) =
dU | det(1 − U )|2iωo .
(4.6)
U (N)
Here µ is the “chemical potential” defined in (3.9) and ωc,o are the energies of the closed and open string vertex operators. Now consider the integrated approach. Following (3.12), the integrated amplitude is Aint 2 (ωc , ω0 ) = −iπ
(2πλ)−i(ωc +ωo ) B2 (ωc , ωo ) . sinh(π(ωc + ωo )
(4.7)
Here B2 is defined by “analytic continuation" from imaginary momenta. Setting ωc,o = inc,o gives B2 (inc , ino ) = IN (ino ) ,
(4.8)
with N = nc + no . Below we will evaluate Aint 2 using both the integrated expression (4.7) and the perturbative expansion (4.4) and show that they agree. This bulk-boundary amplitude captures the main features of more general amplitudes, but in a cleaner context. It allows one to vary the initial state of the brane by creating an open string perturbation. We can therefore explore a key physical question: are the general features of brane decay sensitive to the precise initial state introduced by Sen. For these reasons, we turn to a detailed study of the bulk-boundary two point function. 4.1. Use of the Selberg integral. Our basic technical goal is to evaluate integrals of the form (4.6). This can be done with the help of a famous integral due to Selberg. (For a pedagogical review, see [32].) Following Sect. 2.1 of [33], 1 dtN iti dt1 dU | det(1 − U )|−2α = |e − eitj |2 |1 − e−iti |−2α ··· N! 2π 2π =
N 2 −2αN
2 N!(2π)N ×
i<j
∞ −∞
···
∞ −∞
i
dx1 · · · dxN
|xi − xj |2
i<j
N (1 + xl2 )−N +α .
(4.9)
l=1
The Selberg integral of interest to us is [32] ∞ N dx1 · · · dxN |xi − xj |2 (1 + xl2 )−N+α −∞
=
i<j
(2π)N N ! 2
N 2 −2αN
l=1
N
(j )(j − 2α)
j =1
((j − α))2
.
We thus find that the integral relevant for the open-closed amplitude is N (j )(j − 2α) −2α IN (iα) = dU | det(1 − U )| = . ((j − α))2 j =1
(4.10)
(4.11)
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Selberg’s integral converges for real α when α < 21 , which is also the condition for the Gamma functions on the right side to have positive arguments. We will define the result for arbitrary α by analytic continuation from the convergent region. In order to define the integrated amplitudes (4.7) for general real momenta we will have to also analytically continue N . The latter seems particularly problematic since the left-hand side of (4.11) is an integral over U (N ) matrices and the right side contains a discrete product involving N . Happily, progress can be made using the integral representation of the Gamma function, ∞ dt e−t − e−zt −t log (z) = , (4.12) (z − 1)e − t 1 − e−t 0 which is valid for z > 0. (This domain of validity is the same as that required for the convergence of the Selberg integral (4.10), namely α < 1/2 for real α.) Applying this identity gives the result:
∞
log [IN (iα)] = log [B2 (i(N − α), iα)] = 0
dt e−Nt − 1 (1 − eαt )2 . (4.13) t 2(1 − cosh t)
Both α and N can be analytically continued in this expression. We will use this below to compute and compare the perturbative (4.5) and integrated (4.7) 2-point scattering amplitudes. Although the integral expression is itself only convergent in some regions of the complex momentum plane, we will see that it can be computed exactly in terms of special functions in these regions which can then be continued to general values of the momenta.
4.2. Computing the Bulk-boundary amplitude. • Integrated approach: In the approach in which the zero mode is integrated at the outset we wish to evaluate (4.7). We can do this by analytically continuing (4.13) as α → −iωo and N → −i(ωc + ωo ). This gives: (2πλ)−i(ωc +ωo ) ∞ dt H (t,ωo ) (ei(ωc +ωo )t −1) , e0 sinh (π(ωc + ωo )) (1 − e−iωo t )2 . H (t, ωo ) ≡ 2t (1 − cosh t)
Aint 2 (ωc , ωo ) = −iπ
(4.14)
We will evaluate the integral in the exponent in closed form in terms of special functions below. But first let us compare (4.14) to the perturbative result (4.5). • Perturbative calculation: In the perturbative approach we must calculate the sum in (4.5). In terms of the function H (t, ωo ) defined above, this becomes F (0, ωc , 1, ω0 ; µ) =
∞
e−Nµ e
∞ 0
dt H (t,ωo )(e−tN −1)
N=0
l ∞ e−Nµ ∞ = dta H (ta , ωo ) (e−Nta − 1) . (4.15) l! 0 l,N=0
a=1
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We can now expand out the product of (e−ta N − 1) factors in a series and explicitly carry out the sum on N . This gives l ∞ (−1)l−m l l! m
F (0, ωc , 1, ωo ; µ) =
l=0 m=0
∞ 0
l
dta H (ta , ωo )
a=1
1 1 + 2π λ ex
m
0−
b=1 tb
,
(4.16)
where we have substituted back for the chemical potential µ in terms of the coupling λ and the zero mode x 0 . The full amplitude (4.4) involves a Fourier transform of x 0 as in (4.4). We recognize the required transform from the expression (3.19) used for the computation of the closed string one-point function and find ∞ l (2πλ)−i(ωc +ωo ) (−1)l−m l A2 (ωc , ωo ) = −iπ sinh(π(ωc + ωo )) l! m ×
l=0 m=0
∞
0
l
dta H (ta , ωo ) ei(ωc +ωo )
m
b=1 tb
.
(4.17)
a=1
Carrying out the sums we arrive at the final result A2 (ωc , ωo ) = −iπ
(2πλ)−i(ωc +ωo ) ∞ dt H (t,ωo )(ei(ωc +ωo )t −1) . e0 sinh(π(ωc + ωo ))
(4.18)
This precisely matches the integrated amplitude (4.14).7 For future purposes, we define G(ωc , ωo ) ≡
∞
dt H (t, ωo )(ei(ωc +ωo )t − 1)
(4.19)
0
as the exponent in the bulk-boundary amplitude. The fact that the integrated and the perturbative approaches match exactly greatly increases our confidence in the methods developed here. The integrated approach is based on defining the expressions in (3.12) by first expanding them for integer momenta, evaluating the integrals and then interpreting the result as definition of the amplitude for general momenta. Such an “analytic continuation" is inherently problematic – without further specification of asymptotic conditions there exist many analytic functions which coincide with the one at hand at integer momenta. By contrast, the perturbative approach does not suffer from this failing and thus the agreement between (4.18) and (4.14) is very encouraging. Nevertheless, both approaches used here involve analytic continuation in the momenta and run the risk of missing non-analytic pieces. In particular, the perturbative approach assumed a well-defined Fourier transform and thus could miss non-analytic contributions to the amplitude associated with zero momentum processes. Despite this we will be able to extract useful insights into the structure of D-brane decay from (4.18). 7
Related results in the Liouville approach appear in [19].
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4.3. Comparison with other approaches. As discussed in the previous section, one might have proposed an alternative approach to evaluating these amplitudes, by analytically continuing ω2 to imaginary integer values (−iω2 = n ∈ Z + ), since in that case [16] shows that 2 dU | det(1 − zU )|−2n = (1 − z¯z)−n . (4.20) U (N)
This expression is valid for n and N nonnegative integers with n ≤ N . Except for the trivial case N = 0, the region where this formula is valid is disjoint from the region α < 21 , where the Selberg integral was valid prior to analytic continuation in α. To compare the two approaches, set α = n ∈ Z + in (4.11) and take z = 1 in (4.20) to get the proposed answer for the bulk-boundary amplitude. We find the following two possible results: N (j )(j −2n) j =1 ((j −n))2 = finite , −2n dU | det(1 − U )| (4.21) = 2 (0)−n = ∞ . The first expression is obtained from the Selberg formula used in (4.11) while the second arises from (4.20) as derived in [16]. The Selberg result is finite for N > 2n and vanishes for N ≤ 2n. By contrast, the second formula diverges for any N . The consequence of this is that after continuation back to real momenta, naive use of the latter expression from [16] leads to a vanishing amplitude for all momenta, while the Selberg formula will lead to a finite result. This situation can be given a useful worldsheet interpretation. When a bulk vertex operator is taken towards the boundary to define a bulk-boundary amplitude, it will collide with operators from the boundary interaction. Therefore, we need to add counterterms to operators which are taken to the boundary in order to dress them appropriately in the presence of the interaction. The second formula in (4.21) corresponds, however, to a bare vertex operator, and so diverges. On the other hand, the analytic continuation used to define the Selberg integral is a convenient way of regulating and renormalizing the operator, and so gives a finite result. Of the two results displayed in (4.21), the finite Selberg result is the physically relevant one. The summation over N that was carried out in the perturbative approach, and which led to (4.18) for the final amplitude, became possible because the Fourier transform in the definition of A2 (4.4) simplified the expressions. However, it can also be of interest to do the sum on N in (4.5) without integrating over the zero mode x 0 . As we discussed earlier, the contributions to the scattering amplitude from late times (large x 0 ) could shed light on the nature of the “tachyon matter” state to which the brane is supposed to decay. Doing the perturbative sum in (4.5) exactly for fixed small x 0 and then continuing to large x 0 turns out to be very difficult for arbitrary momenta. However, it is possible to make progress for special imaginary integer momenta. The relevant techniques and results are presented in Appendix A. 4.4. Initial state perturbations and the decay of the brane. Our result (4.14) (or (4.18)) for closed string emission from a decaying brane perturbed by an additional open string tachyon allows us to explore how the emission of closed strings is modified when the initial conditions for the brane are changed. For example, we might make a tachyon lump on the brane by superposing open string vertex operators and ask how this affects the decay.
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Important questions to answer are whether all the energy in the brane decays into closed strings, and what is the distribution of these strings. Sen has argued that brane decay leads to a pressureless state of “tachyon matter”[4]. A computation of closed string emission from decaying branes by [8] (summarized in Eq. (3.19) in Sect. 3) showed that at tree level the emission of closed strings is exponentially suppressed in their energy. For low dimension branes this nevertheless implies a divergent total emission, because of the Hagedorn growth in the density of states. This divergence was interpreted as indicating that all the energy of the decaying brane would be converted into closed strings, and that very heavy strings would dominate the decay products. Such heavy strings would have the stress tensor of pressureless dust, thus suggesting that they constitute the mysterious tachyon matter. It was also argued that near the endpoint of the decay, including back-reaction would ensure that energy conservation was satisfied so that the emission of closed strings shuts off after all the energy in the original brane has been converted. This was confirmed in the c = 1 matrix model, where it was also found that it is crucial to take into account the quantum mechanical nature of the decaying brane [12]. By contrast higher dimensional branes had a finite total emission of energy. The conventional wisdom states that in this case, small perturbations can also lead to decay into higher co-dimension branes [6] which would account for the “missing energy” in the decay into closed strings. Our results can enable a systematic exploration of how changing the brane initial conditions affects the decay products. To initiate this study we will compute a closed form expression for the closed string emission amplitude (4.14) in terms of special functions and then extract the asymptotics of the decay for large closed string energies. Before plunging into the calculation, let us note that there are two possibilities for what one might mean by “changing the brane initial conditions”. One interpretation involves describing the brane by a boundary state; then to change the boundary state we can act with an exponentiated open string vertex operator (at least to first order in the operator; beyond that there will be corrections). This corresponds to changing the classical open string tachyon profile of the decaying brane. To first order in the perturbation, the total emission rate into closed strings will then be the sum of the original rate plus a correction due to the perturbation. Here we will focus on an alternative interpretation, in which we add a single quantum open string excitation on top of the classical background. The open-closed amplitude then corresponds to the amplitude for the incoming open string to disappear and for a closed string to be created. To compute the bulk-boundary amplitude (4.14) in closed form we must evaluate an integral of the form G(ωc , ωo ) = 0
∞
dt e−βt − 1 (1 − e−iωo t )2 2t (1 − cosh t)
;
β = −i(ωc + ωo ) .
(4.22)
Notice first that this integral vanishes if ωo = 0, i.e., in the absence of any open string perturbation. In that case, A2 (ωc , 0) = −iπ
(2πλ)−iωc , sinh(π ωc )
(4.23)
reproducing the known result in [8]. So the full result has the form of a modulation of the emission amplitude without an open string perturbation by a factor of eG . It is also easy to argue that the real part of G will be negative so that the modulating factor suppresses the amplitude.
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The challenge in evaluating the integral (4.22) occurs because as t → 0, cancellations between the terms in the numerator are needed to cancel the overall power of t 3 in the denominator in this limit, leading to a finite integral. With this in mind, to evaluate G for general momenta, observe that G also vanishes if β = i(ωc + ωo ) = 0. Because of this, we can write
β
G(ωc , ωo ) =
dβ 0
∂G ≡− ∂β ωo
β
∞
dβ 0
0
dt e−βt (1 − e−iωo t )2 .(4.24) 2 (1 − cosh t)
After expanding (1 − eiωo t )2 , each term in ∂G/∂β can be evaluated with the help of the integral formula
∞
1 − cosh t
ν
e−γ t dt = eiπν 2−ν
0
(γ − ν)(2ν + 1) ; (γ + ν + 1)
1 Reν > − , (4.25) 2
which is listed in [34] (Vol. 1, p. 163). This expression diverges as ν → −1, but the linear combination −e−iπν 2−ν−1 (2ν + 1)
(γ − ν) (γ − ν + iω0 ) −2 (γ + ν + 1) (γ + ν + 1 + iωo ) (γ − ν + 2iωo ) + (γ + ν + 1 + 2iωo )
(4.26)
needed here is well-defined as ν → −1. The limit ν → −1 gives ∂G/∂β in terms of derivatives of the Gamma function. The integral with respect to β that remains can be evaluated by again using the integral representation (4.13). The result can be written in terms of the Hurwitz ζ function 1 ζ (s, z) = (s) and ζ m,n (s, z) ≡
0
∞ t s−1 e−zt
1 − e−t
=
∞ n=0
1 , (n + z)s
(4.27)
∂ m+n ζ (s,z) ∂s m ∂zn . In order to summarize the results, we introduce the notation
H[F (x), a, b)] ≡ F (b + a) − 2F (b) + F (b − a) .
(4.28)
Then, after using various ζ function identities, we find the result: G(ωc , ωo ) = H[ζ (1,0) (−1, x), −iωo , iωo ] − H[ζ (1,0) (−1, x), −iωo , −iωc ] +H[x(ζ (0, x) − ln (x)), −iωo , iωo ] −H[x(ζ (0, x) − ln (x)), −iωo , −iωc ] . (4.29) Putting this expression back into (4.18) gives the general bulk-boundary amplitude in closed form in terms of zeta functions. This result can now be used to extract many aspects of the physics of decaying branes. Here we will explore one question – how does including an open string perturbation change the asymptotics of brane decay into closed strings?
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To do this we must find the asymptotics of G(ωc , ωo ) for ωc ωo . We will use the expressions 1 −x , 2 ∞ B2 (a) + a B2 a −k+2 Bk B2 (a) ln a ζ (1,0) (−1, a) = − + − . (4.30) 2 4 4 (k − 2)(k − 1)k ζ (0, x) =
k=3
Here B2 (a) = a 2 − a + 1/6 and B2 = 1/6 are the second Bernoulli polynomial and number respectively. The second formula was derived for (a) → ∞ [35], but it will apply by analytic continuation for large |a| in general. Applying these formulae we find that for ωc ωo , the dominant term is G(ωc , ωo ) ≈ −ωo2 ln ωc . If ωo is itself large the formula is ω c G(ωc , ωo ) ≈ −ωo2 ln . (4.31) ωo For small ωo and ωc ωo the denominator in the log is modified, but the −ωo2 ln ωc behaviour always holds. The next to leading terms are independent of ωc , but dependent on ωo and include a phase. Beyond that terms are suppressed by powers of ωc . We have checked these asymptotics numerically. To summarize, the amplitude for emitting high energy closed strings, when the brane initial state has been perturbed by a reasonably energetic open string which is subsequently absorbed, is A2 (ωc , ωo ) ≈ −iπ
(2πλ)−i(ωc +ωo ) ωo ωo2 ··· , sinh(π(ωc + ωo )) ωc
(4.32)
where the ellipses represent terms that are either constant or fall off as ωc increases. For the unperturbed initial state of the D-brane (the spatially homogeneous decay), it was found in [8] that the total energy of closed strings emitted was divergent for a Dp brane with p ≤ 2 (but see also [36]). We now examine how this statement is modified when the initial state is perturbed by addition of the boundary tachyon vertex operator. To compute the expectation value of total emitted energy we add up the squares of the individual emission amplitudes, 1 (s) E ∼ |A2 (ωc , ωo )|2 . Vp 2 s
(4.33)
|A2 (ωc , ω0 )| ∼ e−2πωc ωc −ω0 .
(4.34)
For large ωc , 2
−ωo2
This differs from the unperturbed case in the extra factor of ωc to the one done in [8] yields E 1 − p2 −ωo2 dω ∼ ω . c c Vp (2π)p −ω2
. A similar calculation
(4.35)
Notice that the extra factor of ωc o is a suppression if ωo is real but an enhancement for imaginary ωo . Without the initial state perturbation, the average energy per unit volume
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diverges for the D0, D1, and D2 branes. For the D0 brane, the on shell condition fixes 0 the initial state perturbation with the open-string tachyon vertex operator to be eX , i.e. ωo = i and the perturbation enhances the closed string emission leading to a divergent result. For the D1 brane, the result depends on the details of the perturbation. If the spatial momentum is large enough, such that ω0 > 1, the closed string emission will be finite. For higher dimensional branes, the total emission will be finite for any perturbation with real ω0 . If the spatial momentum on any brane is sufficiently low, the on-shell condition will cause ωo to be imaginary, and there will be an enhancement of closed string emission. For p ≤ 3 the energy contained in closed string emission will diverge in this case. 4.5. Comments. For the D0 brane, the divergent closed string emission is expected to be an artifact of perturbation theory, which will be regulated by back reaction and quantum effects. It strongly suggests that the initial D-brane decays completely into closed strings and that tachyon matter, the end product of the decay in classical open string theory, is made up of very heavy closed strings [8]. A similar interpretation is given to the divergent emission of closed strings from the unperturbed D1 and D2 branes. For higher dimensional branes (p ≥ 3) the conventional wisdom has been that the decay process is more elaborate since it proceeds inhomogeneously ([6, 37, 38]). Notice, however, that in a quantum mechanical treatment of a decaying brane of any dimension, there will always be a component of the brane wavefunction along operators with imaginary 0 ωo . This is because such an open string operator will be proportional to e|ωo |X and so be very small at early times, in effect corresponding to an infinitesimal perturbation of the initial state. We have shown that such perturbations enhance closed string emission for all Dp-branes, even when the decay is homogeneous, and would lead to divergent emission (without treating backreaction and wavefunction effects) for p ≤ 3. In fact, we will get a divergent emission for any p if we are willing to consider non-normalizable perturbations with imaginary spatial momenta. This enhancement of closed string emission is appealing. It suggests that a full treatment of decaying branes of any dimension will show all the energy of the brane entering into a state of heavy closed strings. In our study of the open-closed amplitude we have discussed the open string state as an in-state, or a perturbation of the initial condition, while the closed string is emitted as an out-state. By complex conjugating we could have naively written down an amplitude that would apparently describe the emission of an open string. Likewise, at the perturbative level, we could have computed amplitudes containing additional strings in the final state including, apparently, open strings. This seems puzzling given that the brane has decayed at late times and thus open strings should not exist. It is possible that such amplitudes are simply inconsistent at a non-perturbative level. However, it seems more reasonable that we should interpret such amplitudes with “open strings” in the final state as follows. In any time dependent background we construct vertex operators at early and late times (in- and out- operators) to describe simple perturbations around the respective vacua. However, we are always free to use the complete basis of in-operators to describe outgoing states also – these operators will simply describe very complicated out-states. In our context, both open and closed strings can exist at early times, but only closed strings exist at late times. An out-open-string operator should thus represent some very complicated correlated state of closed strings that emerges from the open string via the decay of the brane. This interpretation leads to the important question of whether, in order to compute the full decay of a brane, we must calculate amplitudes with all possible out vertex operators, or whether computing the emission of closed strings as in [8], or as done above, suffices. In other words, do we include all decay channels that
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are perturbatively apparent including the ones with final-state “open strings”, or is this “double counting”? The answer to this question goes to the heart of the open-closed duality for decaying branes that has been conjectured by Sen ([10, 14]). Regardless, the amplitudes computed above are essential components of the full answer and are the complete result for the “exclusive” amplitude describing absorption of the initial open string and emission of the final closed string. 5. The Bulk-Bulk Amplitude Now we turn to a computation of the bulk-bulk amplitude which describes the scattering of a closed string from the decaying brane, or the emission/absorption of a correlated pair of closed strings during the decay. To obtain the relevant correlator from (3.7) and (3.8) we use the conformal symmetry (but recall footnote 6) to locate one vertex operator at z = 0 and the other at z = r, 0 ≤ r ≤ 1 on the unit disk. This gives 2 0 A2 (ω1 , ω2 , r) = |r|−ω1 ω2 (1 − |r|2 )−ω2 /2 dx 0 eix (ω1 +ω2 ) F (0, ω1 , r, ω2 ; µ) , (5.1) with µ = −x 0 − ln(−2πλ) and F (0, ω1 , r, ω2 ; µ) =
∞
e−Nµ JN (ω2 , r) ,
N=0
JN (ω2 , r) =
dU | det(1 − rU )|2iω2 .
(5.2)
U (N)
To compute the full amplitude we should integrate over r as
1 0
dr r.
5.1. The Toeplitz determinant. The key steps in computing the amplitude are to evaluate the terms JN in the infinite series (5.2) and carry out the sum. Each JN is an expectation value of a periodic function with respect to the circular unitary ensemble (Appendix B): π n dta 1 JN (ω2 , r) = |1 − re−ita |2iω2 |eita − eitb |2 N ! −π a 2π a=1
≡ EN
N
a
f (ti ) .
(5.3)
i=1
By Heine’s identity [39], this expectation value is equal to the Toeplitz determinant of the Fourier coefficients of f , EN
n
f (ti ) = DN [f ] ≡ det(fˆj −k )1≤j,k≤N .
(5.4)
i=1
By this notation we mean the determinant of a matrix in which the entry in the j th row and k th column is the (j − k)th Fourier coefficient of f . For reference, the proof of this identity is included in Appendix B. In our case the Fourier coefficients are (denoting α = −iω2 ) dt r |k| |1 − re−it |−2α e−ikt = F (α, |k|+α, |k|+1; r 2 ). (5.5) fˆk = fˆ|k| = 2π |k|B(α, |k|)
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Here B is a beta function and F is a hypergeometric function. This integral representation of the hypergeometric function is valid for α = 0, −1, −2 · · · and |r| < 1 [40]. As we will see below, the boundary limit r → 1 of the right-hand side will be well-defined anyway. The restriction on α can also be written as ω2 = 0, −i, −2i, −3i · · · . • Special imaginary momenta: When ω2 = i, 2i, 3i · · · (i.e., positive imaginary integer momenta), or equivalently α = 1, 2, 3 · · · with α ≤ N we know that the integral (5.2) could have been done by the group theoretic methods of [16] with the result that 2
JN (ω2 , r) = (1 − r 2 )ω2 ;
ω2 = in, n ∈ Z + ; n ≤ N .
(5.6)
Since these momenta are within the range of validity of the integral representation (5.5), the determinant DN [f ] in (5.4) must reproduce (5.6) at positive imaginary integer ω2 . It is easy to verify that this is indeed the case, at least for small n and N but we have not constructed a general proof. • Boundary limit gives Selberg: As r 2 → 1, the integrals and coefficients (5.4) reduce to (4.11) of the previous bulk-boundary calculation. In the limit, the determinants (5.4) reduce to (1 − 2α) N (α + |i − j |) DN [f ]r=1 = det . (5.7) ((1 − α))(α) (1 + |i − j | − α) i,j =1,... ,N Hence this determinant must be equal to (4.11): JN (ω2 , 1) = DN [f ]r=1 =
N (j )(j − 2α) . ((j − α))2
(5.8)
j =1
We have checked the identity explicitly with Maple for N = 1, . . . , 8, but have not constructed a general proof. These two special limits appear initially to be in contradiction with each other since, for ω2 = in, (5.6) diverges as r → 1, while (5.8) is finite. (Also see the discussion around (4.21).) However, what we are really seeing here is that the order of limits can matter when one approaches the boundary at r → 1 at special momenta. Our Toeplitz determinant (5.4) provides a definition of the two point amplitude for general momenta and gives a well-defined analytic continuation to the bulk-boundary amplitudes (with r = 1) that we derived in the previous section. In addition the determinant reproduces the special form (5.6) which was previously obtained at imaginary integer moment [16]. Therefore, it appears that naive analytic continuation of (5.6) to general ω2 , which is the technique used in the past, is invalid, as one might have expected. Below we will see that the naive continuation of (5.6) from imaginary integers to general ω2 amounts to a large N approximation. 5.2. Large N approximation. In general, dealing with the Toeplitz determinants is very complicated. We will consider two methods. For large N , approximate results can be found by using Szeg¨o’s limit theorem. This estimate can be improved by using an identity due to Borodin and Okounkov [41] (see also [42]). The latter authors considered Toeplitz determinants Dn [f ] of periodic functions and related them to Fredholm determinants of certain kernels. In particular, they considered functions of type
f = (1 + ξ1 η)z (1 + ξ2 η−1 )z ,
(5.9)
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where ξ1 , ξ2 , z, z are complex parameters and η is a coordinate on a complex plane. If we set η = −e−it , ξ1 = ξ2 = r, and z = z = −α, we recover our particular case. Here α can be an arbitrary complex number. Following [41], there is an identity DN [f ] = Z det(1 − KN ) .
(5.10)
The prefactor Z involves the Fourier coefficients of ln f : ∞
Z = exp
k(ln f )k (ln f )−k .
(5.11)
k=1
This is the large N approximation for the determinant given by Szeg¨o’s limit theorem, limN →∞ DN [f ] = Z, whereas the factor det(1 − KN ) contains all the corrections to the approximation. To evaluate Z we use the relation [40] (assuming 0 ≤ r 2 ≤ 1) π α r |k| (ln f )k = − ln(1 − 2r cos(t) + r 2 )e−ikt dt = −α , (5.12) 2π −π |k| which gives DN [f ] ≈ Z = exp{α 2
∞
k r 2k k −2 } = (1 − r 2 )−α . 2
(5.13)
k=1
This large N approximation reproduces the naive analytic continuation n2 → α 2 of (3.16) which results from calculations at imaginary integer momenta [16]. In fact, as we showed in (5.6), when α = n, n ∈ Z + , n ≤ N , the result (5.13) is exact, or equivalently, the Borodin-Okounkov determinant det(1 − KN ) equals 1. There appears to be a remarkable localization phenomenon in the amplitude (5.2) such that for special momenta the result localizes to the large N result Z (5.11). Notice that the right-hand side of (5.13) is independent of the dimension N of the matrix. It is dangerous to use this approximate large N result in the bulk-bulk amplitude (5.2) because of the infinite sum, but if we do so regardless, we get (restoring α = −iω2 ) F (0, ω1 , r, ω2 ; µ) =
∞
2
e−Nµ (1 − r 2 )ω2 = 2
N=0
(1 − r 2 )ω2 1 + 2π λex
0
(5.14)
.
Putting this back in the complete amplitude (5.1) and integrating over x 0 gives A2 (ω1 , ω2 , r) = −iπ
(2πλ)−i(ω1 +ω2 ) 2 |r|−ω1 ω2 (1 − |r|2 )ω2 /2 . sinh(π(ω1 + ω2 ))
(5.15)
This gives the same result as in (3.18), computed by doing the zero mode integral first and using naive continuation from large imaginary integer momenta. The full amplitude is computed by including the spatial part of the amplitude and integrating over the position of the vertex operator. To do this recall that the spatial part of the conformal field is unchanged by the boundary perturbation, so we can include the contributions for a standard D-brane in (2.11), and write 1 A2 = dr r A2 (ω1 , ω2 , r) 0
= −iπ
(2πλ)−i(ω1 +ω2 ) sinh(π(ω1 + ω2 ))
1 0
2
dr r |r|t (1 − |r|2 )s/2−2+ω2 ,
(5.16)
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where we used the Mandelstam variables (2.2) and the closed string tachyon on-shell condition ((2.3) with N = 0). Finally, integrating over r gives the result A2 = −iπ
2 (2πλ)−i(ω1 +ω2 ) ((t/2) + 1)(k|| − 1) , sinh(π(ω1 + ω2 )) ((t/2) + k||2 )
(5.17)
where k|| is the spacelike part of the momentum in the directions parallel to the brane. The factor (t/2 + 1) in this expression contains the closed string poles (2.3). The remaining poles coming from the (k||2 − 1) are more exotic, since they occur at special spacelike momenta. If momentum here was not simply the spacelike piece, we would be recovering open string poles. Above we pointed out that the boundary limit r → 1 of the full amplitude (5.8) gives the result of the Selberg integral that we used to compute the bulk-boundary amplitudes in Sect. 4. By contrast, the large N approximation described here, since it coincides with the naive continuation from imaginary integer momenta, gives a singular boundary limit. (See (4.21) and the discussion around it.) In Sect. 4.3 we argued that the singular behavior of the naive analytic continuation could be related to a need to regulate and renormalize bulk operators as they approach the boundary and collide with the boundary perturbation. Here we have seen that a large N approximation of the bulk result leads to a singular boundary limit, while the full expression leads to the finite quantity that was used in Sect. 4 to study the effect of initial state perturbations on the decay of the brane. It may be useful to think of the large N approximation (5.13) as a saddlepoint contribution, with the Borodin-Okounkov determinant in (5.10) containing corrections due to fluctuations in the interactions between the vertex operators and the boundary perturbations. Perhaps it is possible to understand the determinant in terms of a renormalization or dressing of string vertex operators by the boundary interaction. Our result demonstrates that the usual technique for computing decaying brane amplitudes by analytically continuing from imaginary integer momenta is unreliable. The naive continuation only isolates a large N contribution to the full answer. This happens because of the remarkable localization of the full integral to the large N contribution at imaginary integer momenta. Our amplitudes are directly defined at real momenta and differ in important ways from the naive continuation. We have already discussed the consequences of the extra contributions for open-closed amplitudes. We now turn to a brief examination of the extra factor, the Borodin-Okounkov determinant, in a general amplitude. 5.3. The Borodin-Okounkov determinant. The determinant in the full amplitude DN [f ] = Z det(1 − KN ) can be expressed ([41] and [42]) as a Fredholm determinant of an operator KN , acting on the l2 normed infinite vector space l2 (N, N + 1, . . . ), defined by det(1 − K) = 1 +
∞
(−1)m
m=1
∞
det[K(li , lj )]m i,j =1 .
(5.18)
N≤l1
This equation is essentially a discrete version of the Fredholm determinant of a kernel familiar from the theory of integral equations. For the particular case of (5.9) (with the parameters replaced by ours), the kernel K takes the form (correcting a minor error in [41])
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K(i, j ) =
(−α)i+1 (−α)j +1 i+j +2 (1 − r 2 )−2α−1 r i!j ! 2
F (−α − 1, 1 + α, j + 2; r 2r−1 ) 1 r2 × F (α, −α, i + 1, 2 ) i−j r −1 j +1 2
−
F (−1 − α, 1 + α, i + 2; r 2r−1 ) i+1
F (α, −α, j + 1;
r2 ) . (5.19) r2 − 1
The notation (a)b indicates Pochhammer symbols. From the point of view of the scattering amplitude, an important property of the kernel K is that it contains the information that is missed in the naive analytic continuation of the previous results. As a consistency check, we can examine the behavior of (5.18) as r → 0. From (5.5) we can see that the Toeplitz determinants reduce to 1 as r → 0. In addition, from (5.13) Z → 1 as r → 1, so we expect that det(1 − K) → 1 as well. In the limit the kernel factorizes: K(li , lj ) → r 2 · (−α)li +1 r li (−α)lj +1 r lj .
(5.20)
As r → 0 this of course implies that det(1 − K) → 1. In fact, a stronger statement is true – det K vanishes to leading order in r: 2m det[K(li , lj )]m i,j =1 = r
m
(−α)li r li
i=1
×
(−1)π (−α)lπ(1) r lπ(1) · · · (−α)lπ(m) r lπ(m) = 0 . (5.21)
π∈Sm
Because of this the det(1 − K) factor has no effect on the pole structure of the amplitude at r = 0, in agreement with what we expect from the OPE of the tachyon vertex operators. We already know that in the r → 1 limit the Toeplitz determinant reduces to the form (5.8) and the bulk-bulk amplitude reduces to the bulk-boundary amplitude. Since the expression Z (5.13) is singular as r → 1, there must be a compensating singularity in det(1−K). This is more challenging to isolate. It would be very interesting to understand the properties of the determinant better since many aspects of the physics are clearly stored in it. 5.4. What we learn. We have given a prescription for defining the two point scattering amplitude for decaying branes at general momenta, and demonstrated that the usual tactic of naively continuing from imaginary integer momenta is a large N approximation that misses important contributions. The naive continuation fails because, remarkably, the general amplitude localizes to the large N contribution at these special momenta. Since we are using analytic continuation in our methods it is also possible that non-analytic contributions such as zero-momentum delta functions are missed. These are also essential to understand, because they make contributions to important quantities such as the stress tensor. In any case, it is clear that understanding the properties of Toeplitz determinants emerging from U(N) random matrix theory will lead to progress in the study of decaying branes and vice versa. A useful way of making progress with the bulk-boundary amplitude in Sect. 4 was to do the integral over x 0 first, thus avoiding the need to carry out a perturbative sum over
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N as in (5.2). Instead, the result then depends on a single U (N ) integral with N being related to the total energy in the process. Such an approach might be useful here since for large energies the simple expression (5.13) might suffice. The general amplitude prior to integrating over the time x 0 is: F (0, ω1 , r, ω2 ; µ) =
∞
e−Nµ JN (ω2 , r) ; µ = −x 0 − ln(−2π λ) .
(5.22)
N=0
This has the form of a computation in a grand canonical ensemble of U (N ) matrix models. At very early times (x 0 → −∞) the expression is dominated with smallest matrices. As x 0 increases, larger and larger matrices begin to play an important role. As x 0 increases beyond − ln(2πλ) infinitely big matrices dominate the result and the formal power series fails to converge. To find the contribution to the amplitude from such late times, we must analytically continue the fully summed expression from the convergent region x 0 < − ln(2πλ). In view of the many relationships between large U(N) matrices and closed strings, as well as recent developments in 2d string theory ([11–14]), it is tempting to conclude that as the brane decays and spacetime becomes more closed-string-like in character, large U(N) matrices emerge from the dynamics of the disappearing open strings to describe the physical degrees of freedom of spacetime. 6. Discussion Several interesting questions in the physics of decaying branes, including the nature of the final “tachyon matter” state and the possibility of defining a decoupling limit leading to timelike holography, require a deeper understanding of how strings scatter from such unstable objects. This has been a difficult problem to solve because the decay involves a non-trivial boundary interaction on the open string worldsheet. The standard technique to calculate amplitudes in this system is to first perform the necessary integrals at a special discrete set of momenta and then optimisitically “analytically continue” the resulting formal expressions. Of course this procedure is not strictly well-defined – an analytic function cannot be defined from its values at a discrete set of arguments without further information, such as either the behavior at infinity or consistency conditions derived from symmetry arguments. However, in the absence of further information this is the best we can do. In this paper we have made progress in arriving at a better-defined prescription for computing amplitudes. First we showed that the timelike part of general amplitudes could be written in terms of correlators in U (N ) matrix models. (While we only studied tachyon scattering, vertex operators with more oscillators could be treated in a similar manner.) In a perturbative approach, a grand canonical ensemble of matrix models appeared, while integrating over the zero mode led to a single matrix expectation value. A key point is that the resulting open-closed 2-point amplitude can be exactly computed in an open subset of the complex momentum plane using Selberg’s integral. Analytic continuation from this region to real momenta is reliable. Indeed, we were able to compute the open-closed amplitude in closed form in terms of zeta functions, and we derived the asymptotics of closed string emission when the initial condition for the decay is perturbed by an open string operator. In particular we found that for extended branes the emission of heavy closed strings could be enhanced or suppressed by the presence of the additional vertex operator, with interesting consequences for the structure of the “tachyon matter” state to which the brane decays. We likewise showed that the closed-closed scattering amplitude
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V. Balasubramanian, E. Keski-Vakkuri, P. Kraus, A. Naqvi
can be expressed exactly in terms of certain Toeplitz determinants of hypergeometric functions. The results obtained by naive analytic continuation from integer momenta amount to the leading large N approximation (due to Szeg¨o) of these determinants. In time dependent quantum field theories, different choices of analytic continuation in amplitudes are often related to different choices of vacuum state. We did not address the issue of the vacuum in detail here, but it would interesting to relate choices of analytic continuation to choices in the definition of the boundary state for the decaying brane. (See [8, 43, 19, 20], for some relevant discussion.) In this paper we have focussed on the bulk-bulk and bulk-boundary two point functions, and it is an important problem to study more general correlation functions. For example, it would be interesting to study the boundary-boundary correlation function in depth using the ideas developed here. One would like to make contact with the Liouville based approach in, e.g., ([7, 18–21]). In addition to the techniques described in the main text there are other powerful tools that can be brought to bear on the subject of scattering from decaying branes. In Appendix C we show, by extending old techniques of Douglas [24], that there is a free fermion formulation of scattering from decaying branes. This is related to the free fermion techniques that can be used to study decaying branes in 2d string theory ([11–14]). Appendix D points out the relation between our scattering problem and a grand canonical ensemble for a classical Coulomb gas. It would be very interesting to apply these methods, and the techniques we have developed in the main text, to a study of tachyon matter and to explore potential decoupling limits leading to timelike holography, perhaps in the context of recently discovered non-singular spacetime backgrounds representing decaying branes ([44–47]). Acknowledgement. We are grateful to Jan de Boer, Neil Constable, Eric D’Hoker, Michael Gutperle, Antti Kupiainen, and Volker Schomerus for useful conversations. Some of this work was carried out during the “Time and String Theory” workshop at the Aspen Center for Physics. V.B. was supported by the DOE under grant DE-FG02-95ER40893 and by the NSF under grant PHY-0331728. E.K-V. was supported in part by the Academy of Finland. P.K. was supported in part by NSF grant PHYT-0099590. A.N. was supported by Stiching FOM.
Appendix A. Special Momenta and Summing Over N in the Bulk-Boundary Amplitude At late (early) times x 0 , the chemical potential µ (3.9), is large and negative (positive). Thus at late times, the series expression (4.5) combined with the Selberg result (4.11) for the bulk-boundary amplitude does not converge. To get an explicit function of x 0 we must sum (4.5) at early times and then analytically continue. Such a procedure is rather delicate – terms that are small at early times may dominate after the analytic continuation to late times, so the sum needs to be done exactly, or at least with attention paid to the pieces that dominate late time behavior. Progress can be made for special momenta. We can write the amplitude (4.5) as F (0, ω1 , 1, ω2 ; µ) = 1 + =1+
∞ N=1 ∞ N=1
xN
N (j )(j + 2iω2 ) ((j + iω2 ))2
j =1
x N c(N, ω2 ) ,
(A.1)
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with x = e−µ = −2πλex . The coefficients obey 0
c(N + 1, ω2 ) (N + 1)(N + 1 + 2iω2 ) . = c(N, ω2 ) ((N + 1 + iω2 ))2
(A.2)
A series of this form is, by definition, a hypergeometric function if the ratio (A.2) is a rational function of N . In our case, this happens if iω2 is a positive integer. In fact, when iω2 = m is a non-negative integer a little algebra shows that c(N, −im) is a polynomial in N : m−1 k N (j )(j + 2m) N N N m (1 + )(1 + ) = 1 + . c(N, −im) = ((j + m))2 m k 2m − k j =1
k=1
(A.3) Furthermore c(0, −im) = 1.
N Setting iω2 = m the amplitude becomes F (1, −im; µ) = 1 + ∞ N=1 x c(N, −im) which we can be written ∞
F (0, ω1 , 1, ω2 ; µ) = 1 + c(x
d xN , −im) dx N=1
d x = 1 + c(x , −im) . dx 1−x We thus arrive at
(A.4)
!
"m m−1 d x dx F (0, ω1 , 1, −im; µ) = 1 + 1 + m k=1
k d d x dx x dx x )(1 + ) . × (1 + k 2m − k 1−x
(A.5)
The simplest cases are 1 , 1−x 1 F (0, ω1 , 1, −i; µ) = , (1 − x)2 1 x 3 − 5x 2 + x − 3 F (0, ω1 , 1, −2i; µ) = − . 3 (1 − x)5 F (0, ω1 , 0, 0; µ) =
(A.6)
Note that the late time region is given by limx0 →∞ x = −∞. It follows from (A.5) that lim F (0, ω1 , 1, −im; µ) = 0.
x0 →∞
(A.7)
So we find that for these special values of momenta the bulk-boundary amplitude vanishes at late time. To summarize, (A.5) combined with (4.4) gives the result for the bulk-boundary two-point function at special momenta.
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Appendix B. Heine’s Identity We consider integrals of the form n n dti In = f (ti ) |eiti − eitj |2 , 2π i=1
i=1
(B.1)
i<j
containing a 2π−periodic function f (t). In the Random matrix literature this corresponds to an expectation value of a function ni=1 f (ti ) with respect to the circular unitary ensemble [32]. In our case, f (t) = |1 − ze−it |2ik . A well known identity due to Heine [39], states that the expectation value of a periodic function f (t) with respect to the circular unitary ensemble is equal to the Toeplitz determinant of the Fourier coefficients fˆk = f (t) exp(−ikt)dt, In (B.2) = Dn [f ] = det(fˆ(j −k) )1≤j,k≤n . n! A Toeplitz determinant is a determinant of a (Toeplitz) matrix where all entries are equal along diagonals. For convenience, we review the proof of Heine’s identity [39]. ¯ where is the VanderFirst, note that the i<j |eiti − eitj |2 term is equal to , i(k)t l monde determinant = det{e }k,l=1,···n . The integral can be written as: n n dti ¯ . In = f (ti ) (B.3) 2π i=1
i=1
Now expand the determinants out explicitly, = permutation of integers 1 · · · n. Hence, ¯ =
n−1
(−1)1 +2 e
(−1)
ei
n
i (1 (l)−2 (l))tl
l=1 (l)tl
, where is a
.
(B.4)
1 ,2 l=0
Inserting this in (B.3), we obtain In =
(−1)
1 +2
1 ,2
# n dtl i[(1 (l)−2 (l))tl ] . f (tl ) e 2π
(B.5)
# n dtl f (tl ) eiνl tl . 2π
(B.6)
l=1
Let νl = 1 (l) − 2 (l). Then In =
(−1)1 +2
1 ,2
l=1
We can fix the first permutation 1 = 1, 2, 3, · · · , n and multiply the result by n!. This simplifies the above expression: # n In = n! fˆ(l−(l)) (−1)
l=1
= n! det(fˆ(l−m) )1≤l,m≤n .
(B.7)
The determinant in the final expression is a Toeplitz determinant of the Fourier coefficients of f (t).
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Appendix C. Fermionization In this appendix, we outline a relation between the closed string scattering amplitudes and correlation functions in a theory of free fermions on a circle which gives an interesting reformulation of the scattering problem. The basic idea is due to Douglas [24] who showed the relation between U(N) group theory and a theory of N free fermions on a circle. As we saw in Sect. 3, the scattering amplitudes can be written as a sum of correlators in a grand canonical ensemble of unitary matrix models, with time setting the fugacity: ∞ N F (x0 , z, ωc ) = dMU (N) | det(1 − zM)|2iωc , (−x) (C.1) N=1
where x = written as
2π λex0 .
Using techniques developed in [24], we can show that this can be
F (x, z, E2 ) =
∞
(−x)N N|e
dθ † (θ)f (θ)(θ)
|N ,
(C.2)
N=1
where f (θ) = −µ ln(1+r 2 −2r cos θ ) and |N is the N -fermion vacuum state satisfying Bn |N = 0, † B−n |N = 0,
N −1 , 2 N −1 |n| ≤ . 2 |n| >
(C.3)
Here, we have expanded the fermionic fields (θ ) and † (θ ) in terms of creation and annihilation operators as † Bn einθ , † (θ ) = B−n e−inθ . (C.4) (θ ) = n∈Z
n∈Z
The expression in (C.2) is still quite cumbersome because we have to sum over expectation values in the N fermion ground state. To obtain a more useful expression, we will add a factor of e−βH and sum over all states in the Hilbert space. Then, by projecting onto the N fermion sector and taking the β → ∞ limit correctly reproduces the sum over N fermion ground states. Thus the sum in (C.2) can be written as the β → ∞ limit of the following partition function: † (C.5) Z(β) = Tr e−βH e dθ (θ)f (θ)(θ) (−x)N PN , where PN is the projection operator onto the N particle sector: ˆ PN = dα eiα(N−N) .
(C.6)
Here, Nˆ = dθ † counts the number of fermions. The Hamiltonian H in the N fermion sector is given by 1 d † d N (N 2 − 1) H ≡ H0 + E0 = − , (C.7) dθ 2 dθ dθ 12
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† d N(N 2 −1) where we have defined H0 = 21 dθ d . Notice that the N dθ dθ and E0 = − 12 fermion ground state |N has zero energy. Using these definitions we can write (C.5) as † e−i(α+π)N+βE0 +ln x . (C.8) Z(β) = dαTr e−βH0 e dθ (θ)(f (θ)+iα)(θ) N
To proceed further, we notice that the trace in this expression can be computed by path integral methods: D † De−S O . (C.9) Tr(e−βH O) = (0)=(β)
The path integral naturally computes the expectation values of operators which are normal ordered, which here means that all the † ’s are pushed to the left. Thus to compute the trace † (C.10) Tr e−βH0 e dθ (θ)(f (θ)+iα)(θ)
from a path integral, we write the operator e operator:
e
dθ † (θ)(f (θ)+iα)(θ)
=: e
dθ † (θ)(f (θ)+iα)(θ)
dθ † (θ)e(f (θ )+iα) (θ)
as a normal ordered : .
(C.11)
Then, the trace can be written as a path integral expression as follows: 2 dθ dt † i ∂t∂ + 21 ∂ 2 +e(f (θ )+iα) δ(t) −βH0 † dθ † (θ)(f (θ)+iα)(θ) ∂θ e = D De Tr e ∂ 1 ∂2 = det i + +eiα (1+r 2 −2r cos θ)−µ δ(t) , 2 ∂t 2 ∂θ (C.12) where the path integral is performed over periodic ’s satisfying (0, θ ) = (β, θ ). This determinant is still non-trivial to compute and here we do not push it any further. If this determinant is computed, then the β → ∞ limit of Z(β) in (C.8) can give the amplitude (C.1). Appendix D. Relation to a Grand Canonical Ensemble of a Classical Gas It is interesting to note that the series (5.2) can also be rewritten as the grand canonical partition function of a Coulomb gas of point charges in two dimensions, confined in a circle and interacting with two external point charges via repulsive Coulomb interactions [48]. The point charges on the circle interact via a two body potential V (ti , tj ) = − ln |eiti − eitj |.
(D.1)
The vertex operators at z1 = 0 and z2 = r correspond to two external point charges at 1/z1∗ = ∞ and 1/z2∗ = 1/r of strength −α, acting on the charges on the circle by the potential ln |1 − za eitj | = α ln |1 − reitj |. (D.2) V (tj ) = α a=1,2
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Now, we can introduce a temperature 1/β = 1/2, and the coefficients Jn (5.3) can be interpreted as the partition function of a gas of n charges interacting with the two external ones at temperature T = 1/2:
n
dta ZN (T ) = JN = (D.3) e−β( a=1 V (ta )+ a
If we formally introduce a fugacity z = eµ = −2πλex , the infinite series (5.2) becomes the grand canonical partition function: Q(T , z) = F (r, α; x ) = 0
∞
zN ZN (T ) .
(D.4)
N=0
Note that for λ > 0, the fugacity has a “wrong” negative sign. Further, eventually we will want α = iω2 , so that the strength of the charge at z2 = 1/r becomes imaginary. References 1. Strominger, A.: The dS/CFT correspondence. JHEP 0110, 034 (2001); Inflation and the dS/CFT correspondence. JHEP 0111, 049 (2001) 2. Balasubramanian, V., de Boer, J., Minic, D.: Mass, entropy and holography in asymptotically de Sitter spaces. Phys. Rev. D 65, 123508 (2002); Holography, time and quantum mechanics. Talk presented at the 3rd Sakharov Intl. Conf. on Physics, Moscow, June, 2002, http://arxiv.org/abs/arXiv:gr-qc/0211003, 2002; Exploring De Sitter Space And Holography. Class. Quant. Grav. 19, 5655 (2002) [Ann. Phys. 303, 59 (2003)] 3. Gutperle, M., Strominger, A.: Spacelike branes. JHEP 0204, 018 (2002) 4. Sen, A.: Rolling tachyon. JHEP 0204, 048 (2002) ; Sen, A.: Tachyon matter. JHEP 0207, 065 (2002); Sen, A.: Field theory of tachyon matter. Mod. Phys. Lett. A 17, 1797 (2002) 5. Strominger, A.: Open string creation by S-branes. http://arxiv.org/list/hep-th/0209090, 2002 6. Larsen, F., Naqvi, A., Terashima, S.: Rolling tachyons and decaying branes. JHEP 0302, 039 (2003) 7. Gutperle, M., Strominger, A.: Timelike boundary Liouville theory. Phys. Rev. D 67, 126002 (2003) 8. Lambert, N., Liu, H., Maldacena, J.: Closed strings from decaying D-branes. http://arxiv.org/list/ hep-th/0303139, 2003 9. Sen, A.: Open and closed strings from unstable D-branes. Phys. Rev. D 68, 106003 (2003) 10. Sen, A.: Open-closed duality at tree level. Phys. Rev. Lett. 91, 181601 (2003) 11. McGreevy, J., Verlinde, H.: Strings from tachyons: The c = 1 matrix reloaded. JHEP 0312, 054 (2003) 12. Klebanov, I.R., Maldacena, J., Seiberg, N.: D-brane decay in two-dimensional string theory. JHEP 0307, 045 (2003) 13. Douglas, M.R., Klebanov, I.R., Kutasov, D., Maldacena, J., Martinec, E., Seiberg, N.: A new hat for the c = 1 matrix model. http://arxiv.org/list/hep-th/0307195, 2003 14. Sen, A.: Open-closed duality: Lessons from matrix model. Mod. Phys. Lett. Al 9, 841–854 (2004) 15. Goulian, M., Li, M.: Correlation Functions In Liouville Theory. Phys. Rev. Lett. 66, 2051 (1991) 16. Constable, N.R., Larsen, F.: The rolling tachyon as a matrix model. JHEP 0306, 017 (2003) 17. Teschner, J.: Liouville theory revisited. Class. Quant. Grav. 18, R153 (2001) 18. Strominger, A., Takayanagi, T.: Correlators in timelike bulk Liouville theory. Adv. Theor. Math. Phys. 7, 369 (2003) 19. Schomerus, V.: Rolling tachyons from Liouville theory. JHEP 0311, 043 (2003) 20. Fredenhagen, S., Schomerus, V.: On minisuperspace models of S-branes. JHEP 0312, 003 (2003) 21. Fredenhagen, S., Schomerus, V.: Exact Rolling Tachyons. To appear 22. Gaiotto, D., Itzhaki, N., Rastelli, L.: Closed strings as imaginary D-branes. Nucl. Phys. B 688, 70 (2004) 23. Bergman, O., Razamat, S.S.:Imaginary time D-branes to all orders. JHEP 0406, 046 (2004) 24. Douglas, M.R.: Conformal field theory techniques for large N group theory. http:// arxiv.org/list/hep-th/9303159, 1993 25. Polchinski, J.: String Theory. Cambridge: Cambridge University Press, 1998
394
V. Balasubramanian, E. Keski-Vakkuri, P. Kraus, A. Naqvi
26. Hwang, S.: Cosets as gauge slices in SU(1,1) strings. Phys. Lett. B 276, 451 (1992); Evans, J.M., Gaberdiel, M.R., Perry, M.J.: The no-ghost theorem for AdS(3) and the stringy exclusion principle. Nucl. Phys. B 535, 152 (1998) 27. Okuda, T., Sugimoto, S.: Coupling of rolling tachyon to closed strings. Nucl. Phys. B 647, 101 (2002) 28. Okuyama, K.: Comments on half S-branes. JHEP 0309, 053 (2003) 29. de Boer, J., Sinkovics, A., Verlinde, E., Yee, J.T.: String interactions in c = 1 matrix model. JHEP 0403, 023 (2004) 30. Sen, A.: Rolling tachyon boundary state, conserved charges and two dimensional string theory. JHEP 0405, 076 (2004) 31. Kraus, P., Ryzhov, A., Shigemori, M.: Strings in noncompact spacetimes: Boundary terms and conserved charges. Phys. Rev. D 66, 106001 (2002) 32. Mehta, M.: Random Matrices. London: Academic Press, Second edition, 1991 33. Keating, J.P., Snaith, N.C.: Random Matrix Theory and ζ (1/2 + it). Commun. Math. Phys. 214, 57 (2000) 34. Erdelyi, A. (ed.): Higher Transcendental Functions. Vols. 1-3, Malabar, FL: Kreiger, 1981 35. Weisstein, E.W. et al.: Hurwitz Zeta Function. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HurwitzZetaFunction.html; Also see http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/06/02/ 36. Karczmarek, J.L., Liu, H., Maldacena, J., Strominger, A.: UV finite brane decay. JHEP 0311, 042 (2003) 37. Sen, A.: Time evolution in open string theory. JHEP 0210, 003 (2002) 38. Mukhopadhyay, P., Sen, A.: Decay of unstable D-branes with electric field. JHEP 0211, 047 (2002) 39. Heine, H.: Kugelfunktionen. Berlin, 1878 and 1881, Reprinted W¨urzburg: Physica Verlag, 1961; Szeg¨o, G.: Orthogonal Polynomials. New York: AMS, 1959 40. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Corrected and enlarged edition, London-New York: Academic Press, 1980 41. Borodin, A., Okounkov, A.: A Fredholm determinant formula for Toeplitz determinants. Int. Eqs. Oper. Th. 37, 386–396 (2000) 42. Basor, E.L., Widom, H.: On a Toeplitz determinant identity of Borodin and Okounkov. Int. Eqs. Oper. Th. 37, 397–401 (2000) 43. Maloney, A., Strominger, A., Yin, X.: S-brane thermodynamics. JHEP 0310, 048 (2003) 44. Jones, G., Maloney, A., Strominger, A.: Non-singular solutions for S-branes. Phys. Rev. D 69, 126008 (2004) 45. Wang, J.E.: Twisting S-branes. JHEP 0405, 066 (2004) 46. Tasinato, G., Zavala, I., Burgess, C.P., Quevedo, F.: Regular S-Brane Backgrounds. JHEP 0404, 038 (2004) 47. Lu, H., Pope, C.N., Vazquez-Poritz, J.F.: From AdS black holes to supersymmetric flux-branes. http://arxiv.org/list/hep-th/0307001, 2003; Lu, H., Vazquez-Poritz, J.F.: Four-dimensional Einstein Yang-Mills de Sitter gravity from eleven dimensions. Phys. Lett. B 597, 394 (2004); Lu, H., Vazquez-Poritz, J.F.: Non-singular twisted S-branes from rotating branes. JHEP 0407, 050 (2004) 48. Baker, T.H., Forrester, P.J., Pearce, P.A.: Random matrix ensembles with an effective extensive external charge. http://arxiv.org/list/cond-mat/9803355, 1998 Communicated by M.R. Douglas
Commun. Math. Phys. 257, 395–423 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1326-5
Communications in
Mathematical Physics
A Monomial Basis for the Virasoro Minimal Series M(p, p ) : The Case 1 < p /p < 2 B. Feigin1 , M. Jimbo2 , T. Miwa3 , E. Mukhin4 , Y. Takeyama5 1 2 3 4 5
Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia. E-mail: [email protected] Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan. E-mail: [email protected] Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. E-mail: [email protected] Department of Mathematics, Indiana University-Purdue University-Indianapolis, 402 N.Blackford St., LD 270, Indianapolis, IN 46202, USA. E-mail: [email protected] Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan. E-mail: [email protected]
Received: 24 May 2004 / Accepted: 12 October 2004 Published online: 22 March 2005 – © Springer-Verlag 2005
Abstract: Quadratic relations are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The first case is the (2, 1) primary fields for the (p, p )-minimal series Mr,s (1 ≤ r ≤ p − 1, 1 ≤ s ≤ p − 1) for the Virasoro algebra where 1 < p /p < 2. We restrict ourselves to the case p ≥ 3, for which the (2, 1) primary field exists. The second case is the intertwiners corresponding to the two-dimensional representation for the level k integrable highest weight modules V (λ) (0 ≤ λ ≤ k) for the affine Lie algebra sl2 . 1. Introduction In this paper, we study quadratic relations satisfied by intertwiners (or primary fields) in chiral conformal field theory in two basic cases: (I) the (p, p )-minimal series Mr,s (1 ≤ r ≤ p − 1, 1 ≤ s ≤ p − 1) for the Virasoro algebra where 1 < t = p /p < 2 and p ≥ 3, and (II) the level k integrable highest weight modules V (λ)(0 ≤ λ ≤ k) for the affine Lie algebra sl2 . We consider special intertwiners: the (2, 1)-primary field acting as Mr,s → Mr±1,s for (I), and the C2 intertwiner acting as V (λ) → V (λ ± 1) for (II). We write explicitly the quadratic relations for these intertwiners. Our aim is to construct bases of representation spaces by using the intertwiners. We construct vectors in representation spaces from the highest weight vectors by the action of monomials in p−1 the Fourier components of intertwiners. In Case (I), the space ⊕r=1 Mr,s is generated from the highest weight vector |b(s), s ∈ Mb(s),s by the action of the (2, 1)-primary field. We choose 1 ≤ b(s) ≤ p − 1 in such a way that the conformal dimension r,s (see (2.1)) of the space Mr,s attains the minimum at r = b(s). Similarly, in Case (II), the space ⊕kλ=0 V (λ) is generated from the highest weight vector |0 ∈ V (0) by the action of the C2 intertwiner. We call these vectors monomials. Using the quadratic relations, we rewrite an arbitrary monomial as a linear combination of monomials satisfying certain admissibility conditions. We then prove that admissible monomials are linearly independent by computing the characters of representations.
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There exist several constructions of monomial bases in terms of the chiral conformal algebras (the Virasoro algebra, the affine Lie algebras, and so on), instead of the intertwiners [3–6]. For comparison we review one such constructions for the Virasoro algebra. In [15] a monomial base of the irreducible module M1,s ( M1,2k+3−s ) (1 ≤ s ≤ k + 1) in the (2, 2k + 3)-minimal series is constructed. In addition to the Lie algebra relations, which is quadratic, the Virasoro generators Ln satisfy additional relations of order k + 1: for each m ∈ Z an element of the form k
l=0
n1 ≤···≤nl n1 +···+nl =m
cn(m) L · · · Lnl 1 ,... ,nl n1
(1.1)
acts as zero on all modules M1,s . These relations are obtained from a single relation for the Virasoro current. Using the relations one can reduce any monomial of the form L−n1 · · · L−nN |1, s (n1 ≥ · · · ≥ nN > 0)
(1.2)
to those which satisfy the difference two conditions ni − ni+k ≥ 2
(1 ≤ i ≤ N − k).
(1.3)
In the case of intertwiners, the operators act from one irreducible module to another. Since different modules have different conformal dimensions, a natural parametrization of the Fourier components is by rational numbers instead of integers. In Case (I), the (2,1) primary field φ (r,r ) (z) acts from Mr,s to Mr ,s , where r = r ± 1. We have the expansion φ (r,r ) (z) = φn z−n−2,1 . (1.4) n∈Z+r ,s −r,s (r,r )
The operator φn
maps Mr ,s → Mr,s . Monomials are of the form (r ,r )
(r
φ−n0 1 1 · · · φ−nNN−1
,rN )
|b(s), s,
(1.5)
where rN = b(s), ri = ri+1 ± 1 and ni ∈ ri−1 ,s − ri ,s + Z. In order to label monomials we need to specify a sequence of integers (r0 , r1 , . . . , rL ), which appear in the composition of intertwiners. We call it a path. The difference two conditions change to ni − ni+1 ≥ w(ri , ri+1 , ri+2 )
(1 ≤ i ≤ N − 1),
(1.6)
where w(r, r , r ) (r = r ± 1, r = r ± 1) are rational numbers. They are given in (3.3–3.5). The distance k and the gap 2 in (1.3) change to the distance 1 and the gap w. In particular, the gap is dependent on the path. Let us illustrate the gap condition in the simplest case w(r + 2, r + 1, r) = t/2. The quadratic relation reads −t/2
z1
(1 − z2 /z1 )−t/2 φ (r+2,r+1) (z1 )φ (r+1,r) (z2 ) −t/2
= z2 (1 − z1 /z2 )−t/2 φ (r+2,r+1) (z2 )φ (r+1,r) (z1 ). n Let (1 − z)−t/2 = n≥0 cn z be the Taylor expansion. In terms of the Fourier coeffcients we have the following relations:
A Monomial Basis for M(p, p )
397
(r+2,r+1) (r+1,r) φn+1 + · · · (r+2,r+1) (r+1,r) (r+2,r+1) (r+1,r) φm+t/2 + c1 φn−t/2−1 φm+t/2+1 φn−t/2
(r+2,r+1) (r+1,r) φm φn + c1 φm−1
=
+ ··· .
(r+2,r+1) (r+1,r)
φn by this relation to a If n − m < t/2, we can reduce a monomial φm (r+2,r+1) (r+1,r) φn , where n − m > n − m. The case linear combination of monomials φm w(r, r ± 1, r) is more involved. Instead of the power function (1 − z2 /z1 )−t/2 we need a hypergeometric series. The details are given in the main text. The combinatorial structure of monomial bases for Case (II) is similar. In this case, the intertwiner has two components corresponding to two weight vectors in the sl2 module C2 . Therefore, in addtion to paths which describe a sequence of level k irreducible modules, we need another kind of paths which describe a sequence of the C2 components. In [13] and [11], respectively, a monomial base in terms of the Ck+1 intertwiner for the level k integrable highest weight sl2 modules, and one in terms of the (2, 1)-primary field in the (3, p ) Virasoro minimal series, respectively, is constructed. To be precise, monomial bases constructed in these papers are not bases for the modules but ones for direct sums of the tensor products of the modules with certain bosonic Fock spaces. The intertwiners are modified by bosonic vertex operators so that they constitute a new VOA. In these cases, paths which describe a sequence of representation spaces are redundant because the target space of the intertwiner is uniquely determined by the initial space. Finally, we mention [14], in which, in the framework of VOA, an existence of an infinite set of quadratic relations is shown by using infinitely many intertwiners. In the present paper, in Cases (I) and (II), we give explicit forms of quadratic relations. We use finitely many intertwiners, only one in Case (I) and two in Case (II), and write finitely many quadratic relations. Each relation consists of infinitely many relations for the Fourier components of the intertwiners. We also prove that the left ideal which annihilates the highest weight vector is generated by these relations (and the highest weight conditions) by showing that monomial bases are obtained by reduction using these quadratic relations. The plan of the paper is as follows. In Sect. 2 we derive the quadratic relations of the intertwiners for Case (I). In Sect. 3 we construct monomial bases for Case (I). Construction of the quadratic relations and the monomial bases for Case (II) is given in Sect. 4. 2. Quadratic Relations for the (2, 1)-Primary Field 2.1. The (2, 1)-primary field in the minimal series. In this subsection we introduce our notation, and summarize some basic facts concerning representations of the Virasoro algebra which will be used in subsequent sections. For these and related formulas, we find the textbook [1] to be a useful reference. Let Vir be the Virasoro algebra with the standard C-basis {Ln }n∈Z and c, satisfying [Lm , Ln ] = (m − n)Lm+n +
c m(m2 − 1)δm+n,0 , 12
Fix a pair (p, p ) of relatively prime positive integers. We set t=
p . p
[c, Ln ] = 0.
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Though we will consider only the case 1 < t < 2 later on, we do not make this assumption in Sect. 2. For each (p, p ), there exists a family of irreducible Vir -modules Mr,s = Mr,s (p, p ) (1 ≤ r ≤ p − 1, 1 ≤ s ≤ p − 1) on which c acts as the scalar 1 cp,p = 13 − 6 t + . t We use the notation |v to represent an element of Mr,s . The module Mr,s is Q-graded with respect to the degree operator L0 , Mr,s = ⊕d∈Z≥0 +r,s (Mr,s )d , (Mr,s )d = {|v ∈ Mr,s | L0 |v = d|v}, where dimC (Mr,s )r,s = 1 and r,s =
(rt − s)2 − (t − 1)2 . 4t
We fix a generator of (Mr,s )r,s and denote it by |r, s. We have Ln (Mr,s )d ⊂ (Mr,s )d−n , so that Ln |r, s = 0 (n > 0), and L−n (n > 0) are the creation operators. ∗ ∗ ∗ = ⊕ Consider the right Vir -module Mr,s d∈Z≥0 +r,s (Mr,s )d , (Mr,s )d = Hom C ∗ (Mr,s )d , C . For u| ∈ (Mr,s )d and |v ∈ (Mr,s )d , we set degu| = d, deg |v = d, ∗ ×M and write the dual coupling Mr,s r,s → C as (u|, |v) → u|v. We denote by ∗ r, s| ∈ (Mr,s )r,s the element such that r, s|r, s = 1. In conformal field theory, the notion of primary fields plays a key role. The (k, l)primary field is a collection of generating series (r ,s ;r,s) (r ,s ;r,s) φk,l (z) = z−n−k,l , φk,l n∈Z+r,s −r ,s
n
(r ,s ;r,s)
)n : Mr,s → Mr ,s . Up to a scalar whose coefficients are linear operators (φk,l multiple, it is characterized by the following properties: (r ,s ;r,s) (r ,s ;r,s) [Ln , φk,l (z)] = zn z∂ + (n + 1)k,l φk,l (z), (2.1) (r ,s ;r,s)
(φk,l
)n (Mr,s )d ⊂ (Mr ,s )d−n .
Here and after, we set ∂ = d/dz. In this paper we will consider only the (2, 1)-primary field assuming that 2 < p < p and p ≥ 3. It exists for (r , s ) = (r ± 1, s) with 1 ≤ r, r ± 1 ≤ p − 1, 1 ≤ s ≤ p − 1. Normally we suppress the index (r, s) and write it as (r±1,s;r,s) φ ± (z) = φ2,1 (z) = φn± z−n−2,1 , (2.2) n∈Z+r,s −r±1,s
where 2,1 = (3t − 2)/4. We choose the normalization 1 ± φ |r, s = |r ± 1, s. r,s −r±1,s
(2.3)
1 Our normalization (2.3) is different from the one commonly used in conformal field theory, where one demands r, s|φ σ (z1 )φ −σ (z2 )|r, s = (z1 − z2 )−22,1 + · · · as z1 → z2 .
A Monomial Basis for M(p, p )
399
It is known that the highest-to-highest matrix element H (z1 , z2 ) = r , s|φ σ1 (z1 )φ σ2 (z2 )|r, s
(r = r + σ1 + σ2 )
satisfies the second order linear differential equation 1 2 1 2,1 22,1 + r,s − r ,s 1 r,s ∂ + + + ∂− 2 − H (1, z) = 0. t z z−1 z (z − 1)2 z(z − 1) The existence, and the above differential equation, are the only information we will need about φ ± (z). Together with the homogeneity H (kz1 , kz2 ) = k r ,s −r,s −22,1 H (z1 , z2 ) and the normalization (2.3), H (z1 , z2 ) are determined as follows: r ± 2, s|φ ± (z1 )φ ± (z2 )|r, s = (z1 z2 )−(t−1)/2±(rt−s)/2 (z1 − z2 )t/2 , ± r, s|φ ∓ (z1 )φ ± (z2 )|r, s = (z1 z2 )−2,1 yr,s (z2 /z1 ), where ± yr,s (z) = zt/4±(rt−s)/2 (1 − z)1−3t/2 F 1 − t, 1 − t ± (rt − s), 1 ± (rt − s); z (2.4) and F (a, b, c; z) denotes the hypergeometric function. In the particular case r = 1 or r = p − 1, there are only the following possibilities: + r = 1 : y1,s (z) = z3t/4−s/2 (1 − z)1−3t/2 F (1 − t, 1 − s, 1 − s + t; z),
− r = p − 1 : yp−1,s (z) = z3t/4−(p −s)/2 (1 − z)1−3t/2
×F (1 − t, 1 − p + s, 1 − p + s + t; z). The relevant hypergeometric functions are reciprocal polynomials, i.e., F (a, −n, 1 − a − n; z−1 ) = z−n F (a, −n, 1 − a − n; z)
(n ∈ Z≥0 ).
(2.5)
From these results and the intertwining property (2.1), it follows that the general matrix elements u|φ σ1 (z1 )φ σ2 (z2 )|v (u| ∈ Mr∗ ,s , |v ∈ Mr,s ) are Laurent series convergent in the domain |z1 | > |z2 |, multiplied by an overall rational power of z1 , z2 . 2.2. A bilinear relation for hypergeometric functions. The hypergeometric equation for ± (z) is invariant under the substitution z → z−1 . In this subsection, we derive an yr,s identity for solutions to such an equation. Let us consider a Fuchsian linear differential equation with regular singularities at 0, 1, ∞, which is also invariant under the substitution z → z−1 . It takes the general form d 2y + dz2
1−µ−ν 1 − λ + − λ− + z z−1
dy + dz
λ + λ− µν + 2 z z(z − 1)2
where λ± , µ, ν ∈ C are parameters satisfying the relation 2(λ+ + λ− ) + µ + ν = 1.
y = 0,
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B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
The corresponding Riemann scheme is 0 1 ∞ λ + µ λ+ . λ ν λ − −
(2.6)
We assume that λ+ − λ− , µ − ν ∈ Z. Then a basis of solutions is given by y ± (z) = C ± (−z)λ± (1 − z)µ F λ+ + λ− + µ, 2λ± + µ, 1 ± (λ+ − λ− ); z with C ± = 0. Fixing the branch by arg (−z) = 0 for z < 0, we have the transformation law y τ (z−1 ) = y σ (z)Bστ , σ =±
where the connection matrix B = ± B± =
(Bστ )σ,τ =±
sin π(λ+ + λ− + µ) , sin π(λ∓ − λ± )
± = B∓
is given by
(λ± − λ∓ ) (1 + λ± − λ∓ ) C ± .
(2λ± + µ) (1 − 2λ∓ − µ) C ∓
Consider now the Riemann scheme 1 ∞ 0 − λ+ −µ −λ+ . − λ 2 − ν −λ − −
(2.7)
(2.8)
Let
yˇ ± (z) = Cˇ ± (−z)−λ± (1 − z)−µ F −λ+ − λ− − µ, − 2λ± − µ, 1 ∓ (λ+ − λ− ); z (2.9)
be a basis of solutions of the corresponding differential equation. From (2.7) it follows that, if C + Cˇ + = −
2λ− + µ − ˇ − C C , 2λ+ + µ
(2.10)
then the connection matrix associated with (2.9) is given by t B −1 . We take Cˇ ± C ± =
2λ∓ + µ , 2(λ∓ − λ± )
(2.11)
so that (2.10) and C + Cˇ + + C − Cˇ − = 1 hold. Lemma 2.1. With the choice (2.11), the following identities hold: y + (z)yˇ + (z) + y − (z)yˇ − (z) = 1, µ 1+z y + (z)(z∂ yˇ + )(z) + y − (z)(z∂ yˇ − )(z) = . 2 1−z
(2.12) (2.13)
Proof. Denote by ϕ(z), ϕ(z) ˜ the left-hand sides of (2.12), (2.13), respectively. They are single valued and holomorphic at z = 0. From the relations of the connection matrices ˜ −1 ) = −ϕ(z). ˜ Therefore they are single mentioned above, we have ϕ(z−1 ) = ϕ(z), ϕ(z 1 valued also at z = ∞, and hence on P . From the Riemann schemes (2.6), (2.8), we see that ϕ(z) is regular at z = 1, while ϕ(z) ˜ has at most a simple pole there. The lemma follows from these facts.
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2.3. Exchange relations. The aim of this subsection is to derive the quadratic exchange relations for the (2, 1)-primary field. σ , g σ , hσ which enter these relations. They First let us introduce several functions fr,s r,s r,s β are of the form z1α z2 ψ(z2 /z1 ), where α, β ∈ Q and ψ(z) is a power series convergent in |z| < 1. Suppose 2 ≤ r ≤ p − 2. Set λ± = r±1,s − r,s = µ = −22,1 = 1 −
t rt − s ± , 4 2
3t , 2
ν = 3,1 − 22,1 =
t , 2
and C ± = eπiλ± . ± (z) in (2.4). The corresponding With this choice of parameters, y ± (z) becomes yr,s ± ± (z). functions yˇ (z) defined by (2.9) and (2.11) will be denoted by yˇr,s Define ± ± fr,s (z1 , z2 ) = (z1 z2 )2,1 z1−1 (1 − z2 /z1 )−1 yˇr,s (z2 /z1 ), ± ± gr,s (z1 , z2 ) = (z1 z2 )2,1 (z∂ yˇr,s )(z2 /z1 ). σ (z , z ) = g σ (z , z ) = 0 except for the In the case r = 1 or p − 1, we define fr,s 1 2 r,s 1 2 following ones: + (z1 , z2 ) = K1−t,s−1 z1 f1,s − fp−1,s (z1 , z2 ) =
3(t−1)/2+s/2−1 (s−1)/2 z2 (1 − z2 /z1 )−2+3t/2 , (2.14) 3(t−1)/2+(p −s)/2−1 (p −s−1)/2 K1−t,p −s−1 z1 z2 (1 − z2 /z1 )−2+3t/2 .
Here Ka,n = F (a, −n, 1 − a − n; 1)−1 =
n−1
j =0 (a
+ j )/(2a + j ). Define also −t/2
(t−1)/2∓(rt−s)/2 h± z1 r,s (z1 , z2 ) = (z1 z2 )
for all r, s. Consider the formal series Fr,s (z1 , z2 ) :=
(1 − z2 /z1 )−t/2
(2.15)
σ fr,s (z1 , z2 )φ −σ (z1 )φ σ (z2 ),
σ =±
Gr,s (z1 , z2 ) :=
σ gr,s (z1 , z2 )φ −σ (z1 )φ σ (z2 ),
σ =± σ (z1 , z2 ) := hσr,s (z1 , z2 )φ σ (z1 )φ σ (z2 ). Hr,s These series have the form m,n∈Z Om,n z1m z2n , where each coefficient Om,n is a well defined linear operator.
Lemma 2.2. The following identities hold: 1 Pr,s (z1 , z2 ), z1 − z 2 z1 + z 2 r, s|Gr,s (z1 , z2 )|r, s = (−2,1 ) (2 ≤ r ≤ p − 2), z1 − z 2 r + 2σ, s|Hσ (z1 , z2 )|r, s = 1, r, s|Fr,s (z1 , z2 )|r, s =
(2.16) (2.17) (2.18)
402
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
where Pr,s (z1 , z2 ) is a homogeneous symmetric polynomial satisfying (2 ≤ r ≤ p − 2), 1 Pr,s (z, z) = zs−1 (r = 1), zp −s−1 (r = p − 1). Proof. Using Lemma 2.1, formulas (2.16), (2.17) for 2 ≤ r ≤ p − 2 and (2.18) are easily verified with Pr,s (z1 , z2 ) = 1. Formula (2.16) for r = 1 follows from P1,s (z1 , z2 ) = z1s−1
F (1 − t, 1 − s, 1 − s + t; z2 /z1 ) F (1 − t, 1 − s, 1 − s + t; 1)
and (2.5). The case r = p − 1 is similar.
Denote by C[z1±1 , z2±1 ]S2 the space of symmetric Laurent polynomials in z1 , z2 . Lemma 2.3. Set Ln = z1n+1 ∂1 + z2n+1 ∂2 + (n + 1)(z1n + z2n )2,1 , ∂j = ∂/∂zj . Then there exist an , bn , cn , dn , en ∈ C[z1±1 , z2±1 ]S2 such that [Ln , Fr,s (z1 , z2 )] = Ln + an (z1 , z2 ) Fr,s (z1 , z2 ) + bn (z1 , z2 )Gr,s (z1 , z2 ), (2.19) [Ln , Gr,s (z1 , z2 )] = cn (z1 , z2 )Fr,s (z1 , z2 ) + Ln + dn (z1 , z2 ) Gr,s (z1 , z2 ), (2.20) σ σ [Ln , Hr,s (z1 , z2 )] = Ln + en (z1 , z2 ) Hr,s (z1 , z2 ). (2.21) Proof. Let us show (2.19) assuming 1 < r < p − 1. Using [Ln , φ −σ (z1 )φ σ (z2 )] = Ln φ −σ (z1 )φ σ (z2 ), we obtain [Ln , Fr,s (z1 , z2 )] − Ln Fr,s (z1 , z2 ) σ = − (z1n+1 ∂1 + z2n+1 ∂2 )fr,s (z1 , z2 ) · φ −σ (z1 )φ σ (z2 ) σ =±
= an (z1 , z2 )Fr,s (z1 , z2 ) + bn (z1 , z2 )Gr,s (z1 , z2 ), where z1n+1 − z2n+1 − 2,1 (z1n + z2n ), z1 − z 2 zn − z2n bn (z1 , z2 ) = 1 . z1 − z 2
an (z1 , z2 ) =
Hence we have (2.19). For r = 1 or p − 1, we find [Ln , Fr,s (z1 , z2 )] − Ln Fr,s (z1 , z2 ) = an (z1 , z2 )Fr,s (z1 , z2 ), where an (z1 , z2 ) = −
zn+1 − z2n+1 s − 1 n (z1 + z2n ) − 22,1 − 1 1 2 z1 − z 2
and s = s (r = 1) or p − s (r = p − 1).
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403
Likewise we have [Ln , Gr,s (z1 , z2 )] − Ln Gr,s (z1 , z2 ) = − 2,1 (z1n + z2n )Gr,s (z1 , z2 ) σ +(z1 z2 )2,1 (z1n − z2n ) (z∂)2 yˇr,s (z2 /z1 ) · φ −σ (z1 )φ σ (z2 ). σ =±
With the substitution (z∂) yˇr,s 2
t z+1 = z∂ yˇr,s − 2z−1
t2 (rt − s)2 3t t z − − ( − 1)( − 2) 16 4 2 2 (z − 1)2
yˇr,s ,
the right-hand side becomes cn (z1 , z2 )Fr,s (z1 , z2 ) + dn (z1 , z2 )Gr,s (z1 , z2 ), with z1n − z2n t 2 3t (rt − s)2 t 2 cn (z1 , z2 ) = − − (z1 − z2 ) − ( − 1)( − 2)z1 z2 , z1 − z 2 16 4 2 2 n n t z1 − z 2 dn (z1 , z2 ) = − (z1 + z2 ) + 2,1 (z1n + z2n ) . 2 z1 − z 2 This proves (2.20). The case (2.21) is similar.
Proposition 2.4. Set δ(z2 /z1 ) = n∈Z z1−n z2n . We have the identities of formal series
σ fr,s (z1 , z2 )φ −σ (z1 )φ σ (z2 ) +
σ fr,s (z2 , z1 )φ −σ (z2 )φ σ (z1 )
(2.22)
(2 ≤ r ≤ p − 2), 1 s−1 −1 (r = 1), = z1 δ(z2 /z1 ) × z1 p −s−1 (r = p − 1), z1 σ −σ σ σ gr,s (z1 , z2 )φ (z1 )φ (z2 ) + gr,s (z2 , z1 )φ −σ (z2 )φ σ (z1 )
(2.23)
σ =±
σ =±
σ =±
σ =±
= (−22,1 )δ(z2 /z1 ) σ hr,s (z1 , z2 )φ σ (z1 )φ σ (z2 ) =
(2 ≤ r ≤ p − 2), hσr,s (z2 , z1 )φ σ (z2 )φ σ (z1 ).
(2.24)
Proof. We show that for any u| ∈ Mr∗ ,s , |v ∈ Mr,s we have modulo (z1 − z2 )C[z1±1 , z2±1 ]S2 , 1 u|v Pr,s (z1 , z2 ), z1 − z 2 z1 + z 2 u|Gr,s (z1 , z2 )|v ≡ (−2,1 ) u|v (2 ≤ r ≤ p − 2), z1 − z 2
u|Fr,s (z1 , z2 )|v ≡
where Pr,s (z1 , z2 ) is as in Lemma 2.2.
(2.25) (2.26)
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B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
We also show that modulo C[z1±1 , z2±1 ]S2 , σ (z1 , z2 )|v ≡ 0. u|Hr,s
(2.27)
The left-hand sides are convergent in the domain |z1 | > |z2 |. Noting that 1 Pr,s (z1 , z2 ) + (z1 ↔ z2 ) = z1−1 δ(z2 /z1 ) × Pr,s (z1 , z1 ), z1 − z 2 we obtain the proposition. By Lemma 2.2, (2.25)–(2.27) hold true in the case u| = r , s|, |v = |r, s. By induction, suppose they are true for u |, |v with degu | + deg |v < degu| + deg |v. We may assume either u| = u |Ln or |v = L−n |v with some u |, |v and n > 0. From Lemma 2.3, we have in the first case with 1 < r < p − 1, u |Ln Fr,s (z1 , z2 )|v ≡ u |Fr,s (z1 , z2 )Ln |v + (Ln + an (z1 , z2 ))u |Fr,s (z1 , z2 )|v +bn (z1 , z2 )u |Gr,s (z1 , z2 )|v 1 ≡ u |Ln |v Pr,s (z1 , z2 ) z1 − z 2 1 z1 + z 2 u |v. + (Ln + an (z1 , z2 )) + bn (z1 , z2 )(−2,1 ) z1 − z 2 z1 − z 2 In the last line we used the induction hypothesis. The second term in the right-hand side is a Laurent polynomial. Indeed, using the expressions for an , bn given in the proof of Lemma 2.3, we find 1 z1 + z 2 + bn (z1 , z2 )(−2,1 ) (z1 − z2 ) (Ln + an (z1 , z2 )) z1 − z 2 z1 − z 2 n − zn z 2 = 2,1 n(z1n + z2n ) − 1 (z1 + z2 ) . z1 − z 2 Since it vanishes at z2 = z1 , the assertion follows. In the same way, in the case r = 1 or p − 1, we are to check that (Ln + an (z1 , z2 )) Pr,s (z1 ,z2 ) is a Laurent polynomial, where an is as in the proof of Lemma 2.3. This can z1 −z2
be verified by noting that (z1n+1 ∂1 + z2n+1 ∂2 )Pr,s (z1 , z1 ) = (s − 1)z1n+s −1 . The other cases can be proved in a similar manner. 3. Monomial Bases in Terms of the Primary Field φ2,1 We fix coprime integers p, p (t = p /p) satisfying 1 < t < 2,
(3.1)
and consider the representations Mr,s (1 ≤ r ≤ p − 1, 1 ≤ s ≤ p − 1) of the Virasoro algebra in the (p, p ) minimal series. We construct a monomial basis of Mr,s by using the (2, 1)-primary field.
A Monomial Basis for M(p, p )
405
3.1. Spanning set of monomials. In this subsection we construct a spanning set of vectors for each Mr,s . In the next subsection, we prove that it constitutes a basis. For each 1 ≤ s ≤ p −1 we define b(s) by the condition that the conformal dimension r,s takes the minimal value at r = b(s) for fixed s. Because of the restriction (3.1), we have [tb(s)] = s or s − 1.
(3.2)
Here [x] is the integer part of x. We construct vectors in the spaces Mr,s (1 ≤ r ≤ p − 1) by applying the Fourier components φn± of the (2, 1)-primary field (2.2) to |b(s), s. To each triple of integers (r, r , r ) satisfying the conditions 1 ≤ r, r , r ≤ p − 1, r = r ± 1 and r = r ± 1 we associate a local weight w(r, r , r ): t , 2 t w(r, r + 1, r) = 2 − + [tr] − tr, 2 t w(r, r − 1, r) = 1 − − [tr] + tr. 2
(3.4)
w(r, r , r ) ≥ 0, w(r, r , r ) ≡ r ,s + r,s − 2r ,s mod Z, w(r, r , r ) = w(p − r, p − r , p − r ).
(3.6) (3.7) (3.8)
w(r, r ± 1, r ± 2) =
(3.3)
(3.5)
We have
A sequence of integers r = (r0 , r1 , . . . , rL ) satisfying the conditions 1 ≤ ri ≤ p − 1 and ri+1 = ri ± 1 is called a one-dimensional configuration of length L. We denote (L) by Ca,c the set of one-dimensional configurations of length L satisfying r0 = a and (L) rL = c. By the definition Ca,c is an empty set unless L ≡ a − c mod 2. Let L ≥ 0 be an integer. We define type s admissible monomials of length L, and associated one-dimensional configurations of length L. A type s monomial of length L is a sequence m = (σ1 , m1 ; . . . , ; σL , mL ) of signs σi = ± (or σi = ±1) and rational numbers mi ∈ Q (1 ≤ i ≤ L). The associated one-dimensional configuration r = r(m) of length L is defined by r(m)L = b(s) and r(m)i−1 = r(m)i + σi
(2 ≤ i ≤ L).
(3.9)
A void sequence is a type s monomial of length 0. We denote it by ∅. The monomial ∅ is the unique type s monomial of length 0. It is an admissible monomial by definition. The associated one-dimensional configuration r(∅) is such that r(∅)0 = b(s). If L ≥ 1, a type s monomial m is admissible if and only if the following conditions, where 1 ≤ i ≤ L − 1, are valid: 1 ≤ r(m)i ≤ p − 1, −mi ∈ r(m)i−1 ,s − r(m)i ,s + Z, −mL ∈ r(m)L−1 ,s − b(s),s + Z≥0 , −mi + mi+1 ∈ w(r(m)i−1 , r(m)i , r(m)i+1 ) + Z≥0 .
(3.10) (3.11) (3.12) (3.13)
With each admissible monomial m = (σ1 , m1 ; . . . , ; σL , mL ) of type s we associate the product of the Fourier components:
406
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama σ1 σL (m) = φm . . . φm , 1 L
(3.14)
and define a vector |m ∈ Mr,s , where r = r(m)0 by |m = (m)|b(s), s.
(3.15)
The degree of the vector |m is given by d(m) = b(s),s −
L
mi .
(3.16)
i=1
This is consistent with the degree d in Mr,s = ⊕d (Mr,s )d given by the operator L0 : (Mr,s )d = {v ∈ Mr,s |L0 v = dv}.
(3.17)
The operator φnσ changes the degree by −n. σi Starting from the vector |b(s), s, we create vectors successively by the operators φm i. σi The conditions (3.10) and (3.11) ensure that the operator φmi is acting as Mr(m)i ,s → σL Mr(m)i−1 ,s , the condition (3.12) must hold if the vector φm L |b(s), s is non-zero, and the condition (3.13) requires that the increment from i + 1 to i of the degree differences σ σi caused by the operators φmi+1 i+1 and φmi is at least w(r(m)i−1 , r(m)i , r(m)i+1 ). We denote by Br,s the set of type s admissible monomials m such that r(m)0 = r. In Br,s the length of elements varies. We will prove Proposition 3.1. The set of vectors {|m|m ∈ Br,s } is a spanning set of vectors in Mr,s . We prepare some notation before starting the proof of Proposition 3.1. We denote by B˜ r,s the set of type s monomials satisfying (3.10), (3.11), (3.12) and r(m)0 = r. We drop the conditions (3.13) from Br,s . The vectors |m are defined for monomials m ∈ B˜ r,s as well. We have Proposition 3.2. The set of vectors {|m|m ∈ B˜ r,s } is a spanning set of vectors in Mr,s . σL σ1 Proof. Let V ⊂ ⊕1≤r≤p−1 Mr,s be the linear span of the vectors φm 1 . . . φmL |b(s), s. The proposition will follow if we show
|r, s ∈ V Ln V ⊂ V
(1 ≤ r ≤ p − 1), (n ∈ Z).
(3.18) (3.19)
Assertion (3.18) is clear from (2.3). Assertion (3.19) for n ≥ 0 follows from the intertwining relation (2.1) and the highest condition Ln |r, s = 0 (n ≥ 0). To verify (3.19) for n < 0, it is enough to show that Ln |b(s), s ∈ V for n = −1, −2. This can be seen by the following formula obtained by using (2.3), (2.1): (r ∓ 1)t − s ± 1 ± φr∓1,s −r,s −1 |r ∓ 1, s, t t L−2 ± L2−1 |r, s (r ∓ 1)t − s ± 1 ± 2 = ± (r ∓ 1)t − s ± 2 φ |r ∓ 1, s. r∓1,s −r,s −2 t L−1 |r, s = ±
The proof is over.
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407
The vectors in Proposition 3.2 are not linearly independent. We want to discard those which are linearly dependent on others. In order to do this procedure systematically, we define a partial ordering of the set B˜ r,s . We will show that if a monomial m ∈ B˜ r,s is non-admissible the vector |m can be written as a linear combination of the vectors associated with monomials smaller than m. Let m, m ∈ B˜ r,s , and let L, L be the lengths of m, m , respectively. We write m < m if and only if L < L , or L = L and there exists 1 ≤ l ≤ L such that mi = mi for all 1 ≤ i ≤ l − 1 and ml < ml . Note that the above ordering is not a total order because we do not compare m, m if they have the same length L and mi = mi for all 1 ≤ i ≤ L, but r(m) = r(m ). Since the number of monomials of a fixed degree is finite, it follows that the vectors associated with non-admissible monomials are contained in the linear span of those associated with admissible monomials. Proposition 3.3. If a monomial m ∈ B˜ r,s does not satisfy (3.13), then it can be written as a linear combination of vectors corresponding to smaller monomials in B˜ r,s . Before starting the proof we prepare some notation and a technical lemma. Recall that ρ = rt − s, λ± = t/4 ± ρ/2. We set δ=
0 if n1 + n2 is even, 1 if n1 + n2 is odd,
w+ = w(r, r + 1, r) = 2 − t/2 + [ρ] − ρ, w− = w(r, r − 1, r) = 1 − t/2 − [ρ] + ρ.
(3.20) (3.21) (3.22)
Note that 1 < w− + w+ < 2. We expand the hypergeometric functions which appear in (2.22): 3t
f ± (z) = (1 − z)−2+ 2 F (−1 + t, −1 + t ∓ ρ, 1 ∓ ρ; z) t
= (1 − z)1− 2 F (2 − t, 2 − t ∓ ρ, 1 ∓ ρ; z) = fn± zn .
(3.23)
n∈Z
For (2.23), we have g ± (z) = (z∂ − λ± ) (1 − z)f ± (z) = gn± zn .
(3.24)
n∈Z
Note that if n < 0, we have fn± = gn± = 0. Lemma 3.4. Define a (2n + δ) × (2n + δ) matrix Mn,δ (ρ)j,k =
+ fj++k−n−1−δ + f−j +k−n if 1 ≤ j ≤ n + δ, + + gj +k−2n−1−δ + g−j +k+1 if n + δ + 1 ≤ j ≤ 2n + δ.
The matrix Mn,δ (ρ) is non-singular.
(3.25)
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B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
Proof. Set Dn,δ (ρ) = det Mn,δ (ρ).
(3.26)
We will show that up to a non-zero constant the determinant factorizes: 2n−1+δ Dn,δ (ρ) = const.
i=1
{(ρ + t − i − 1)(ρ − t − i + 2)} 2n−1+δ (ρ − i)2n−i+δ i=1
n−
i−δ 2
.
(3.27)
Since the determinant never vanishes when 2 ≤ r ≤ p − 2, the proof will be thus finished. Set an (ρ) , i=1 (ρ − i)
fn+ = n
bn (ρ) . i=1 (ρ − i)
gn+ = n
(3.28)
From (3.23) and (3.24) we see that an (ρ) and bn (ρ) are polynomials in ρ of degree n and n + 1, respectively. This implies that if we multiply Dn,δ (ρ) by the denominator in the right-hand side of (3.27), we have a polynomial in ρ of degree 2n(n + δ). Therefore, in order to prove (3.27) it is enough to show that for 1 ≤ i ≤ 2n − 1 + δ by specializing at ρ = −t + i + 1 or ρ = t + i − 2, the corank of the matrix Mn,δ (ρ) becomes at least n − [(i − δ)/2]. We prove the statement above for the case δ = 0 and ρ = −t +i +1. Three other cases are similar. From (3.23) we see that at the special values ρ = −t +i +1 (i ≥ 1) the series t F (z) = (1 − z)−1+ 2 f + (z) becomes a reciprocal polynomial of degree i − 1. Namely, i−1 we have F (z) = j =0 Fj zj , where Fj = Fi−1−j . Suppose that P (z) is a reciprocal polynomial of degree l. Then, the polynomials (1 + z)P (z) and ((1 − z)z∂ + lz) P (z) are also reciprocal and their degrees are l + 1 and l. From this remark and (3.24) it t follows that the series G(z) = (1 − z)−1+ 2 g + (z) is a reciprocal polynomial of degree i. t Let (1−z)−1+ 2 = j ∈Z cj zj be the expansion at z = 0. If we replace fj+ , gj+ by the coefficients Fj , Gj of F (z), G(z), the matrix Mn,δ (ρ) is left-multiplied by C = (Cj k ), where Cj k = ck−j . Therefore, it is enough to prove the statement when the matrix Mn,δ (ρ) is given by Fj , Gj instead of fj+ , gj+ . On closer inspection, the matrix thus obtained proves to have the symmetries (CMn,0 (−t + i + 1))j,k = (CMn,0 (−t + i + 1))j,2n+i−k
(3.29)
for 1 ≤ j, k ≤ n and 1 ≤ 2n + i − k ≤ n. The assertion follows from this. To show that the constant in (3.27) is non-zero, we specialize ρ = −1 and t = 2 which produces a simple matrix with determinant ±2. Proof of Proposition 3.3. We start from Proposition 3.2. Suppose that a monomial m ∈ B˜ r,s does not satisfy (3.13). Then, we have L ≥ 2, and there exists 1 ≤ i ≤ L − 1 such that −mi + mi+1 < w(r(m)i−1 , r(m)i , r(m)i+1 ).
(3.30)
σi σi+1 For the proof of Proposition 3.3 it is enough to show that the vector φm i φmi+1 |v, where σi+2 L |v = φmi+2 · · · φmL |b(s), s can be written as a linear combination of vectors of the form φnσ φnσ |v satisfying σi + σi+1 = σ + σ , mi + mi+1 = n + n and n < mi .
A Monomial Basis for M(p, p )
409 σ
σi i+1 We use the quadratic relations given in Proposition 2.4 to rewrite the product φm i φmi+1 . There are six cases: Case 1 : (r(m)i−1 , r(m)i , r(m)i+1 ) = (r, r + 1, r + 2), Case 2 : (r(m)i−1 , r(m)i , r(m)i+1 ) = (r, r − 1, r − 2), Case 3 : (r(m)i−1 , r(m)i , r(m)i+1 ) = (1, 2, 1), Case 4 : (r(m)i−1 , r(m)i , r(m)i+1 ) = (p − 1, p − 2, p − 1), Case 5 : (r(m)i−1 , r(m)i , r(m)i+1 ) = (r, r + 1, r) (2 ≤ r ≤ p − 2), Case 6 : (r(m)i−1 , r(m)i , r(m)i+1 ) = (r, r − 1, r) (2 ≤ r ≤ p − 2). By the symmetry (r, s) ↔ (p − r, p − s), Case 2 is equivalent to Case 1, and Case 4 is equivalent to Case 3. Case 1 is discussed in the Introduction. We will not repeat the argument. We can ignore the right-hand side of the quadratic identities (2.22) and (2.23), because it contributes with only smaller terms to the corresponding monomials. If we forget the right-hand side, the quadratic relation (2.22) for r = 1 (see (2.14)) is similar to (2.24) (see (2.15)). The difference is in the sign of the second term, and that the power (z1 − z2 )−t/2 in (2.15) is replaced by (z1 − z2 )−2+3t/2 in (2.14). These powers are related to the weight w in (3.13). The effect of the change of sign is that w(1, 2, 1) = 3 − 3t/2 while w(r, r + 1, r + 2) = t/2. Cases 5 and 6 are combined in the relations (2.22) and (2.23), and thus the proof is more involved. By the symmetry (r, s) ↔ (p − r, p − s), without loss of generality we can assume that
ρ = rt − s > 0.
(3.31)
We have the Fourier series expansions (r,s;r±1,s)
φ2,1
(z) =
φn(r,r±1) z−n−2,1 ,
(3.32)
φn(r±1,r) z−n−2,1 .
(3.33)
n∈Z+r±1,s −r,s (r±1,s;r,s)
φ2,1
(z) =
n∈Z−r±1,s +r,s
Here we fix s and suppress it from the notation. Substituting (3.32) and (3.33) in the left-hand side of (2.22) and taking the coefficient of each monomial in z1 , z2 , we obtain (r,r±1) (r±1,r) φn2 . Modulo a conan infinite linear combination of the operator products φn1 (r,r±1) (r±1,r) φn2 . stant in the right-hand side, we obtain a relation among the monomials φn1 Similarly, we obtain another set of relations from (2.23). Each of these relations contains (r,r±1) (r±1,r) φn2 . The sum n1 +n2 is constant for all the infinitely many terms of the form φn1 terms which appear in one relation, and moreover, n2 is bounded from below. Therefore, acting on each vector in Mr,s , only finitely many terms create non-zero vectors. Namely, the infinite quadratic relation for the operators gives a finite linear relation among the vectors created by them. Let us write explicitly the relation. We fix n1 + n2 . In order to simplify the notation, we write (r,r±1) (r±1,r) ± φn2 , n2 −n1 = φn1
(3.34)
since the sum n1 + n2 is fixed, there is no ambiguity when we write only the difference n2 − n1 . Thus, ± m is defined for m ≡ n1 + n2 mod 2.
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For a ∈ Z we set Aa =
± i∈Z
Ba =
± i∈Z
± (fi± + fi−2a−1+δ )± −2a−2λ± +2i+δ ,
(3.35)
± (gi± + gi−2a+δ )± −2a−2λ± +2i+δ .
(3.36)
From (2.22) we obtain the relations Aa ≡ 0 (a ∈ Z≥0 ), and from (2.23) Ba ≡ 0 (a ∈ Z≥δ ). Here ≡ means that we ignore constant terms. In the proof below we always ignore constant terms. In accordance with admissibility of monomials, we say ± n is admissible if and only if n ∈ w± + Z≥0 .
(3.37)
σ n
We say also σn is larger than if and only if n < n (not opposite). Our goal is to show that if σn is non-admissible then it can be written as an infinite linear combination of ± n such that n ≥ w± . Since w± − 1 < w∓ , it implies that the σi σi+1 product φmi φmi+1 can be replaced by a linear combination of φnσ φnσ such that n < mi (modulo a constant). Let us prove the above statement. We set n± (a) = −2a − 2λ± + δ.
(3.38)
The largest term among those ± n which appear in Aa (a ∈ Z≥0 ) or Ba (a ∈ Z≥δ ) is ± n± (a) : Aa = f0+ + n+ (a) + · · ·
+f0− − n− (a) + · · · ,
(3.39)
+g0− − n− (a) + · · · .
(3.40)
Ba = g0+ + n+ (a) + · · · Since the matrix
+ − f0 f0 1 1 = −λ+ −λ− g0+ g0−
(3.41)
σ is non-degenerate, we can replace ± n± (a) by smaller terms n , i.e., n > nσ (a). We do + − replace them while both n+ (a) and n− (a) are non-admissible, i.e., n± (a) < w± . This is equivalent to a ≥ N, where [ρ] 2 + δ if [ρ] is even, (3.42) N = [ρ]+1 if [ρ] is odd. 2
For this value, we have
w+ − 2[ρ] − 2 − δ if [ρ] is even, w+ − 2[ρ] − 3 + δ if [ρ] is odd, w− − 1 − δ if [ρ] is even, n− (N ) = w− − 2 + δ if [ρ] is odd. n+ (N ) =
(3.43) (3.44)
A Monomial Basis for M(p, p )
411
Note, in particular, that n− (N ) < w− and n− (N ) + 2 ≥ w− . Therefore, the non-admissible terms σn with σ = −, and those with σ = + and n ∈ n+ (N ) + 2Z≤0 can be − replaced by the terms + n (n ∈ n+ (N ) + 2Z>0 ) and the admissible terms n (n ≥ w− ). The remaining non-admissible terms are + w+ −2a+δ
(1 ≤ a ≤ 2N − δ) if [ρ] is even,
+ w+ −2a+1−δ
(1 ≤ a ≤ 2N − δ) if [ρ] is odd.
(3.45) (3.46)
We want to eliminate these terms by using the relations Aa ≡ 0 (0 ≤ a ≤ N − 1) and Ba ≡ 0 (δ ≤ a ≤ N − 1). For this we need to show that the (2N − δ) × (2N − δ) matrix whose elements are the coefficients of the non-admissible terms in these relations, is non+ degenerate. If [ρ] = 2n (n ≥ 1), we put the coefficients of + w+ −4N+3δ , . . . , w+ −2+δ in the first,..., the (2N − δ)th column of the matrix, respectively. If [ρ] = 2n − + 1 + 2δ (n ≥ 1 − δ), we put the coefficients of + w+ −4N+1+δ , . . . , w+ −1−δ in the first,..., the (2N − δ)th column of the matrix, respectively. We put the coefficients in A0 , . . . , AN−1 , Bδ , . . . , BN−1 in the first,..., the (2N − δ)th row of the matrix, respectively. If δ is fixed, two matrices corresponding to [ρ] = 2n, 2n − 1 + 2δ are the same. We denote it by Mn,δ (ρ). The size of Mn,δ (ρ) is 2n − δ. This matrix is nothing but Mn,δ (ρ) given by Lemma 3.4. The proof is over. 3.2. One-dimensional configuration sums. In this section we calculate the character of the spanning set Br,s : χ˜ r,s (q) = q d(m) , (3.47) m∈Br,s
where d(m) is given by (3.16). We use (q)L =
L
(1 − q i ),
i=1
and the q-binomials
L M
=
(q)L (q)M (q)L−M
0
if 0 ≤ M ≤ L, otherwise.
The character χ˜ r,s (q) is expressed as follows by using one-dimensional configuration sums: Proposition 3.5.
χ˜ r,s (q) = q b(s),s δr,b(s) + L≥1
1 (q)L
dr = L(rL−1 ,s − b(s),s ) +
L−1 i=1
q dr ,
(3.48)
(L) r∈Cr,b(s)
iw(ri−1 , ri , ri+1 ).
(3.49)
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Proof. The degree of the vector |b(s), s) is b(s),s . The operator product (m) (see (3.14) adds some degree to it. The minimal degree added by those (m) which have the same associated one-dimensional configuration, i.e., r(m) = r, is equal to dr . The fluctuation from the minimum is added up to the factor 1/(q)L . Let a, b, c be integers satisfying 1 ≤ a, b, c ≤ p − 1, b = c ± 1.
(3.50)
We introduce the one-dimensional configuration sum
(L)
Ya,b,c (q) =
(L+1)
r∈Ca,c
q
L
i=1 iw(ri−1 ,ri ,ri+1 )
.
(3.51)
,rL =b
Note that (0)
Ya,b,c (q) = δa,b .
(3.52)
We have χ˜ r,s (q) = δr,b(s) q b(s),s +
q b(s),s (L−1) q L(b(s)+τ,s −b(s),s ) Yr,b(s)+τ,b(s) . (3.53) (q)L
L≥1
τ =±1
Similar sums appeared in the calculation of local height probabilities of the eight vertex solid-on-solid model (see [7, 8]). In fact, there is a connection (first observed in [9]) between one-dimensional configuration sums and characters in conformal field theories. The Virasoro character χr,s in the minimal series for general (p, p ) (i.e., without the restriction p < 2p) is obtained (see (6) in [12]) in the limit L → ∞ of the one-dimensional configuration sum where the local weights are given by 1 , 2 w(r, ˜ r + 1, r) = −[r(p − p)/p ], w(r, ˜ r − 1, r) = [r(p − p)/p ]. w(r, ˜ r ± 1, r ± 2) =
(3.54) (3.55) (3.56)
Set 1 1 g(r) = − (t − 1)r 2 + r. 4 4
(3.57)
If we modify our weight w to w by the gauge transformation w (r, r , r ) = w(r, r , r ) − 1 + g(r) − 2g(r ) + g(r ),
(3.58)
we obtain 1 w (r, r ± 1, r ± 2) = − , 2 w (r, r + 1, r) = [r(p − p)/p], w (r, r − 1, r) = −[r(p − p)/p].
(3.59) (3.60) (3.61)
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413
Since L
iw(ri−1 , ri , ri+1 ) =
i=1
L
iw (ri−1 , ri , ri+1 ) +
i=1
L(L + 1) 2
−g(r0 ) + (L + 1)g(rL ) − Lg(rL+1 ), the gauge transformation does not essentially change the one-dimensional configuration sum. The expressions (3.54) and (3.59) are very similar if we change the sign. However, they are not equal for the same values of p, p . In fact, the way of connecting the onedimensional configuration sums for w˜ to the Virasoro characters is very different from our way of connecting those for w to the same Virasoro characters. The former uses the limit of L, while in our formula the parameter L appears as a summation variable. By a routine calculation (see, e.g., [7]), we have the following result. Proposition 3.6. For integers a, b, c satisfying b = c ± 1, we define L (L) Xa,b,c (q) = q i=1 iw(ri−1 ,ri ,ri+1 ) .
(3.62)
r=(r0 ,... ,rL ,rL+1 ) ri+1 =ri ±1(0≤i≤L) r0 =a,rL =b,rL+1 =c
Then, we have
L
(L)
Xa,b,b±1 (q) = q C(L,a,b,c)
L±(a−b) 2
,
(3.63)
where C(L, a, b, c) =
1 L(L + 1) ∓ L + (t − 1)(a 2 − b2 ) − a + b 4
+L(t − 1)(±2b + 1) + (1 ∓ 2[(t − 1)c])(L ± (a − b)) ,
and (L)
Ya,b,b±1 (q) =
(L) ε Xεa+2np,b,b±1 (q). ε
(3.64)
n∈Z
We omit the proof of this proposition. We only remark that the gauge transformation (3.58) makes the computation shorter, and that the summation (3.64) realizes the boundary conditions (L)
(L)
Ya,0,1 (q) = Ya,p,p−1 (q) = 0 for 1 ≤ a ≤ p − 1.
(3.65)
3.3. Fermionic formulas of Virasoro characters. We denote the corresponding character by χr,s . It is given by the following bosonic formula: ! q r,s pp n2 +(p r−ps)n pp n2 +(p r+ps)n+rs χr,s (q) = . (3.66) q − q (q)∞ n∈Z
We rewrite it by using
n∈Z
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Lemma 3.7. For all integer l, we have q m2 −ml 1 = . (q)∞ (q)m (q)m+l
(3.67)
1 = 0 if m < 0. (q)m
(3.68)
m∈Z
Here, we set
Applying the lemma to the first/second sum in (3.66) by setting l = ∓r + b + 2pn,
(3.69)
and changing the summation over m ∈ Z to the summation over L = 2m + 2pn + l ∈ ∓r + b + 2Z,
(3.70)
we obtain Proposition 3.8. The Virasoro character χr,s can be written as follows where b is any integer: χr,s (q) = q
r,s
L≥0 L+r+b∈2Z
q
L2 −(r−b)2 4
(q)L
n∈Z
qA
L L−r+b −pn 2
− qB
L L−r−b −pn 2
, (3.71)
where A = p(p − p)n2 + {(p − p)r − p(s − b)}n, B = p(p − p)n2 + {(p − p)r + p(s − b)}n + r(s − b). We will identify the formula (3.71), where b = b(s) with (3.53). As a result, we see that if b = b(s), for each L the sum over n in (3.71) is a series with non-negative coefficients. Proposition 3.9. We have the equality χ˜ r,s (q) = χr,s (q).
(3.72)
Proof. We set b = b(s) in (3.71). There are two cases: Case (i) [tb(s)] = s; Case (ii) [tb(s)] = s − 1. In Case (i), we rewrite (3.71) by using the identities L−r+b L−1 L−1 L −pn 2 = q + , (3.73) L−r+b L−r+b L+r−b − pn − pn + pn 2 2 2 L+r+b L L−1 L−1 +pn 2 = L−r−b +q . (3.74) L−r−b L+r+b − pn − pn + pn 2 2 2 For each L ≥ 1, we thus obtain four terms in (3.71). One can check these four terms are equal to the terms in (3.53) and (3.64) corresponding to (ε, τ ) = (1, 1), (1, −1), (−1, −1), (−1, 1), respectively. The proof for Case (ii) is similar. As a corollary, we have Theorem 3.10. The set of vectors |m (m ∈ Br,s ) is a basis of the representation space Mr,s .
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415
2 -Case 4. The sl In this section we construct a basis of the integrable irreducible highest weight modules of the affine Lie algebra sl2 . The construction is quite similar to the case of Virasoro modules. In the sl2 -case we use the vertex operators (see (4.1) below) instead of the (2, 1)-primary fields. We remark that the notations in this section are independent of those in the previous sections. For example, t = p /p is not used; t will be used for an indeterminate in the Laurent polynomial ring C[t, t −1 ]. 4.1. Preliminaries. First we introduce some notation on the affine Lie algebra sl2 and its representations. Denote by E, F and H the generators of sl2 given by 01 00 1 0 E= , F = , H = . 00 10 0 −1 The algebra sl2 is defined by sl2 = sl2 ⊗ C[t, t −1 ] ⊕ CK ⊕ CD. Here K is the central element. We denote the subalgebra sl2 ⊗ C[t, t −1 ] ⊕ CK by sl2 . m Set X(m) := X ⊗ t for X = E, F or H . The commutation relations are given by [X(m), Y (n)] = [X, Y ](m + n) + mδm+n,0 (X|Y )K, [D, X(m)] = mX(m), where (X|Y ) = tr(XY ). Let V (j ) be the (j + 1)-dimensional irreducible module of sl2 . We denote its affini(j ) zation by Vz := V (j ) ⊗ C[z, z−1 ]. In the following, we fix an integer k ≥ 0 and consider highest weight representations of level k. Let λ be an integer such that 0 ≤ λ ≤ k. We denote the integrable irreducible highest weight module of sl2 with the highest weight (k − λ)0 + λ1 by V (λ). Let |λ be the highest weight vector. The Virasoro algebra acts on V (λ) by the Sugawara operator: Ln :=
1 1 : E(n − m)F (m) + F (n − m)E(m) + H (n − m)H (m) : . 2(k + 2) 2 m∈Z
Here : · : is the normal ordering defined by (m < n), X(m)Y (n) 1 : X(m)Y (n) : = (X(m)Y (m) + Y (m)X(m)) (m = n), 2 Y (n)X(m) (m > n). The central charge is given by c =
3k k+2 .
The vector |λ satisfies
L0 |λ = λ |λ,
λ :=
λ(λ + 2) . 4(k + 2)
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The module V (λ) is bi-graded by L0 and H (0). We set V (λ)d,s := {v ∈ V (λ) | L0 = dv, H (0)v = sv}. Then V (λ)λ ,λ = C|λ and V (λ) = ⊕d∈λ +Z≥0 , s∈λ+2Z V (λ)d,s . Consider the dual module V (λ)∗ = ⊕d,s (V (λ)d,s )∗ . It is the right sl2 -module. Let λ| ∈ (V (λ)λ ,λ )∗ be the vector satisfying λ|λ = 1. 4.2. Vertex operators. Set
C((z)) := {
an zn | an ∈ C, an = 0 for n 0}.
n∈Z
The vertex operator φ(z) is a C[z, z−1 ]-linear map (j ) φ(z) = φn z−n−j : V (λ) ⊗ Vz −→ V (µ) ⊗ zµ −λ −j C((z)) n∈λ −µ +Z
which commutes with the action of sl2 and satisfies [Ln , φ(z)] = zn z∂ + (n + 1)j φ(z). (j )
For u ∈ Vz
(4.1)
we define the map
φ(u; z) : V (λ) −→ V (µ) ⊗ zµ −λ −j C((z)),
φ(u; z)v := φ(z)(v ⊗ u).
Then it satisfies [x, φ(u; z)] = φ(xu; z) for
x∈ sl2 .
(4.2)
Consider the function (j )
(j )
µ|φ(z1 )φ(z2 )|λ : Vz2 2 ⊗ Vz1 1 −→ C. Note that the relation (4.1) with n = 0 implies φ(tz) = t −j · t L0 φ(z)t −L0 . Hence we have µ −λ −j1 −j2
µ|φ(z1 )φ(z2 )|λ = z1
µ|φ(1)φ(z2 /z1 )|λ.
The function G(z) := µ|φ(1)φ(z)|λ satisfies the following differential equation: 1 dG 0 1 = + G, κ dz z z−1
(4.3)
A Monomial Basis for M(p, p )
417
where κ=
1 , k+2
λ (1 ⊗ H ), 2
0 = −F ⊗ E − 1 ⊗ F E +
1 1 = E ⊗ F + F ⊗ E + H ⊗ H. 2 Denote by φ ± (z) the vertex operator of the following type: φ ± (z) : V (λ) ⊗ Vz(1) −→ V (λ ± 1) ⊗ zλ±1 −λ −1 C((z)). (1)
We abbreviate V (1) and Vz to V and Vz , respectively. Let {v+ , v− } be a basis of V satisfying H v± = ±v± . We set σ −n−1 φ,n z φσ (z) := φ σ (v ; z) = n∈λ −λ+σ +Z
for σ, = ±. Each Fourier component of φσ (z) gives a map σ : V (λ)d,s −→ V (λ + σ )d−n,s+ . φ,n
We choose the normalization ± φ±, |λ = |λ ± 1. λ −λ±1
(4.4)
Since the vector |λ satisfies F (0)λ+1 |λ = 0, we have
E(−1)k−λ+1 |λ = 0,
1 F (0)|λ + 1 + O(z) , λ+1 1 − φ+ (z)|λ = zλ−1 −λ −1 +1 E(−1)|λ − 1 + O(z) . k−λ+1 + (z)|λ = zλ+1 −λ −1 φ−
(4.5)
We can obtain the following formulae by solving (4.3). Each solution is uniquely determined by (4.4) and (4.5). Proposition 4.1. ± ± λ ± 2|φ± (z1 )φ± (z2 )|λ = (z1 z2 )λ±1 −λ −1 (z1 − z2 ) 2 , κ
− 1 λ+ 1 κ
+ − 2 λ|φ+ (z1 )φ− (z2 )|λ = (z1 z2 )−1 z 2 (1 − z)− 2 × F (−κ, 1 − (λ + 2)κ, 1 − (λ + 1)κ; z), 1 3 3κ − + −1 −1 2 λ+ 2 κ z λ|φ+ (z1 )φ− (z2 )|λ = (z1 z2 ) (1 − z)− 2 λ+1 × F (1 − κ, λκ, 1 + (λ + 1)κ; z), 3κ −1 1− 21 λ+ 21 κ − + −1 λ|φ− (z1 )φ+ (z2 )|λ = (z1 z2 ) (1 − z)− 2 z k−λ+1 × F (1 − κ, 1 − (λ + 2)κ, 2 − (λ + 1)κ; z), 1
− + λ|φ− (z1 )φ+ (z2 )|λ = (z1 z2 )−1 z 2
λ+ 23 κ
3κ
3κ
(1 − z)− 2 F (−κ, λκ, (λ + 1)κ; z).
Here z = z2 /z1 and F (a, b, c; z) is the hypergeometric function.
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4.3. Exchange relations. In this subsection we prove the exchange relations for the vertex operators φσ (z) (σ, = ±). To this aim we prepare the following lemma which plays a similar role to Lemma 2.1. Lemma 4.2. Let F (a, b, c; z) be the hypergeometric function. Set (β−α+1)F (α−1,α+β−1,1+β;z) (1−α−β)F (α−1,α−β−1,1−β;z) A(z) = F (α,α+β,1+β;z)
and B(z) =
F (α,α−β,1−β;z)
β−α
α(β−α) β F (−α,1−α−β,1−β;z) β(1−β) zF (1−α,1−α−β,2−β;z) α F (−α,β−α,β;z) β F (1−α,β−α,1+β;z)
.
Then A(z)B(z) is a polynomial in z of degree one. More explicitly we have (1 + β − 3α) − (α + β − 1)z (1 + β − 3α)z − (α + β − 1) A(z)B(z) = . (4.6) 1 1 Proof. By using the formula F (a, b, c; z) =
we have
A(z) = where
(c) (b − a) (−z)−a F (a, a − c + 1, a − b + 1; z−1 )
(b) (c − a)
(c) (a − b) + (−z)−b F (b, b − c + 1, b − a + 1; z−1 )
(a) (c − b) (|z| > 1, z ∈ R>0 ),
z0 A(z−1 )M(α, β; z), 01
−1
B(z) = M(α, β; z)
B(z
−1
01 ) , 10
sin πα
(1 − β) (−β) (−z)α sin πβ
(1 − α − β) (α − β) . M(α, β; z) =
(β) (1 + β) sin π α (−z)α (−z)α−β
(β − α + 1) (β + α) sin πβ −(−z)α+β
Hence we obtain
A(z)B(z) =
z0 01 −1 −1 A(z )B(z ) . 01 10
This equality implies that A(z)B(z) is a polynomial of degree one.
Proposition 4.3. We have the following identities of formal power series of operators acting on the irreducible highest weight module V (λ) of level k: hσλ (z1 , z2 )φσ1 (z1 )φσ2 (z2 ) − hσλ (z2 , z1 )φσ2 (z2 )φσ1 (z1 ) = 0, (σ = ±), fλσ (z1 , z2 )φσ1 (z1 )φ−σ (z2 ) − fλσ (z2 , z1 )φσ2 (z2 )φ−σ (z1 ) 2 1
(4.8)
= 1 δ1 +2 ,0 δ(z2 /z1 ), gλσ (z1 , z2 )φσ1 (z1 )φ−σ (z ) − gλσ (z2 , z1 )φσ2 (z2 )φ−σ (z1 ) 2 2 1
(4.9)
σ =±
σ =±
= 1 δ1 +2 ,0 z1−1 δ(z2 /z1 ).
(4.7)
σ =±
σ =±
A Monomial Basis for M(p, p )
Here 1 , 2 = ±, δ(z2 /z1 ) =
419
−n n n∈Z z1 z2 − κ2
2 ∓ 2 (λ+1)κ z h± λ (z1 , z2 ) = (z1 z2 ) 1 1
κ
and
· (1 − z)− 2 , κ
fλ± (z1 , z2 ) = Cλ± (z1 z2 )1 1
3κ
1
3κ
× z− 4 ± 2 (λ+1)κ (1 − z)−1+ 2 F (−1 + κ, −1 + κ ± (λ + 1)κ, ×1 ± (λ + 1)κ; z), ± gλ (z1 , z2 ) = C˜ λ± z1−1 (z1 z2 )1 κ
× z− 4 ± 2 (λ+1)κ (1 − z)−1+ 2 F (κ, κ ± (λ + 1)κ, 1 ± (λ + 1)κ; z), κ
where z = z2 /z1 and F (a, b, c; z) is the hypergeometric function. The constants Cλ± , C˜ λ± are given by Cλ+ =
λ(k + λ + 2) , 2k(λ + 1)
Cλ− =
λ−k , 2k
C˜ λ+ =
λ , λ+1
C˜ λ− = −1.
Remark 4.4. In the case of λ = 0, the summation in the left-hand side of (4.8) and (4.9) becomes one term with σ = −. Similarly, if λ = k, it becomes one term with σ = +. Proof of Proposition 4.3. Here we prove the second relation (4.8). The proofs for the other ones are similar. Let ·, · be the bilinear pairing V × V → C defined by v1 , v2 := 1 δ1 +2 ,0 . This pairing satisfies Xu, v + u, Xv = 0,
u, v ∈ V
(4.10)
for X = E, F, H . For ui ∈ V (i = 1, 2) we set fλσ (z1 , z2 )φ σ (u1 ; z1 )φ −σ (u2 ; z2 ) Au1 ,u2 (z1 , z2 ) := σ =±
−
fλσ (z2 , z1 )φ σ (u2 ; z2 )φ −σ (u1 ; z1 ) − u1 , u2 δ(z2 /z1 ).
σ =±
Then (4.8) is equivalent to the equality Au1 ,u2 (z1 , z2 ) = 0. Let us prove it. We show the following identities: λ|Au1 ,u2 (z1 , z2 )|λ = 0, [X(m), Au1 ,u2 (z1 , z2 )] = z1m AXu1 ,u2 (z1 , z2 ) + z2m Au1 ,Xu2 (z1 , z2 ), for X = E, F, H.
(4.11) (4.12)
These identities imply that Au1 ,u2 (z1 , z2 ) = 0 by the same argument as the proof of Proposition 2.4. To prove (4.11) it is enough to consider the case that u1 = v and u2 = v− for = ±. Then (4.11) follows from Proposition 4.1 and the equalities of (1, 1) and (1, 2) elements in (4.6) with α = κ and β = (λ + 1)κ. Let us prove (4.12). From (4.2) we have [X(m), Au1 ,u2 (z1 , z2 )] = z1m AXu1 ,u2 (z1 , z2 ) + Xu1 , u2 δ(z2 /z1 ) + z2m Au1 ,Xu2 (z1 , z2 ) + u1 , Xu2 δ(z2 /z1 ) .
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Note that z1m δ(z2 /z1 ) = z2m δ(z2 /z1 ). Hence (4.12) follows from (4.10). This completes the proof of (4.8). In the proof of (4.9) we should show that ! −σ σ σ σ σ −σ λ| gλ (z1 , z2 )φ (z1 )φ− (z2 ) − gλ (z2 , z1 )φ− (z2 )φ (z1 ) |λ σ =± = z1−1 δ(z2 /z1 ).
σ =±
To prove this, use the equalities of (2, 1) and (2, 2) elements in (4.6).
4.4. Monomial basis. From the exchange relations (4.7), (4.8) and (4.9), we can construct a spanning set of V (µ): Proposition 4.5. The vectors φσ11,n1 · · · φσLL,nL |0,
L ≥ 0, σi ∈ {+, −}, i ∈ {+, −}
(4.13)
satisfying the following condition (4.14) and (4.15) span the irreducible highest weight module V (µ) of level k. Set λL = 0 and λi−1 = λi + σi (i = 1, . . . , L). Then 0 ≤ λi ≤ k, λ0 = µ, ni ∈ λi − λi−1 + Z, (4.14) −nL ≥ 1 , ni+1 − ni ≥ w(λi−1 , λi , λi+1 ) + h(i , i+1 ) (i = 1, . . . , L − 1), (4.15) where the functions w(λi−1 , λi , λi+1 ) and h(i , i+1 ) are defined by 1 w(λ, λ − 1, λ) = (λ + )κ, 2 3 w(λ, λ + 1, λ) = 1 − (λ + )κ, 2 h(+, −) = 1, h(, ) = 0 otherwise.
w(λ ± 1, λ, λ ∓ 1) =
κ , 2
Proof. First we prove that the vectors φσ11,n1 · · · φσLL,nL |0 satisfying only the condition (4.14) span V (µ). Let Wµ be the subspace of V (µ) spanned by the vectors. From (4.4) we have |µ ∈ Wµ . Moreover, from the commutation relation (4.2) and E(0)|0 = F (0)|0 = F (1)|0 = 0,
− E(−1)|0 = kφ+, φ+ |0, 1 −1 +,−1
which follows from (4.5), we see that xWµ ⊂ Wµ for any x ∈ sl2 . Hence Wµ = V (µ). The rest of the proof is similar to that in Sect. 2. We consider quadratic monomials φσ11,n1 φσ22,n2 acting on V (λ), and show that each monomial is reduced to a linear combination of ones satisfying n2 − n1 ≥ w(λ + σ1 + σ2 , λ + σ2 , λ) + h(1 , 2 ) by using the exchange relations in Proposition 4.3. Here we prove it in the case of σ1 = −σ2 and 0 < λ < k. The proofs for the other cases are similar. Fix n1 + n2 and set σ1 ,2 ;n2 −n1 = φσ1 ,n1 φ−σ . 2 ,n2
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We say the monomial σ1 ,2 ;n is admissible if n ≥ w(λ, λ − σ, λ) + h(1 , 2 ). Write down the exchange relations (4.8) and (4.9) in terms of Fourier components. Then we find − − Cλ+ + 1 ,2 ;n + Cλ 1 ,2 ;n−2(λ+1)κ + · · · − − − Cλ+ + 2 ,1 ;−n+(2λ+1)κ + Cλ 2 ,1 ;−n−κ + · · · ≡ 0
(4.16)
and ˜− − C˜ λ+ + 1 ,2 ;n + Cλ 1 ,2 ;n−2(λ+1)κ + · · · ˜− − − C˜ λ+ + 2 ,1 ;−n+(2λ+1)κ+2 + Cλ 2 ,1 ;−n−κ+2 + · · · ≡ 0. Here ≡ means that we ignore the constant terms. From these relations and + − λ C C det ˜ λ+ ˜ λ− = − = 0, Cλ Cλ k
(4.17)
(4.18)
the monomials + 1 ,2 ;n , − 1 ,2 ;n ,
1 n < (λ + )κ = w(λ, λ − 1, λ) and 2 3 n < −(λ + )κ = w(λ, λ + 1, λ) − 1 2
can be written as a linear combination of the rest. Hence it suffices to prove that the monomials − 1 ,2 ;−(λ+3/2)κ ,
+ +−;(λ+1/2)κ
and
− +−;−(λ+3/2)κ+1
can be reduced to admissible ones. Set 1 = 2 = and n = (λ + 1/2)κ in (4.17). Then we have ˜− − C˜ λ+ + ,;(λ+1/2)κ + Cλ ,;−(λ+3/2)κ + · · · ≡ 0. Hence − ±,±;−(λ+3/2)κ can be reduced. Next consider (4.16) and (4.17) with 1 = −2 = and n = (λ + 1/2)κ: + Cλ+ + − ,−;(λ+1/2)κ −,;(λ+1/2)κ − − + Cλ ,−;−(λ+3/2)κ − − −,;−(λ+3/2)κ + · · · ≡ 0
(4.19)
and ˜− − C˜ λ+ + ,−;(λ+1/2)κ + Cλ ,−;−(λ+3/2)κ + · · · ≡ 0. These relations hold for = + and −. Here we note that (4.19) with = + and th e one with = − are equivalent. Hence we have three relations among four monomials
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− + ±,∓;(λ+1/2)κ and ±,∓;−(λ+3/2)κ . The relations are linearly independent from (4.18). By using them we reduce the monomials except + −,+;(λ+1/2) . At last we consider (4.17) with 1 = −2 = and n = (λ + 1/2)κ + 1. Then we find − − C˜ λ− − ,−;−(λ+3/2)κ+1 −,;−(λ+3/2)κ+1 + · · · ≡ 0.
This relation holds for = + and −; however this gives only one relation by the same reason as before. Hence we can reduce one monomial. Here we reduce − +,−;−(λ+3/2)κ+1 . This completes the proof. Theorem 4.6. The vectors (4.13) satisfying (4.14) and (4.15) are linearly independent. Hence they give a basis of the irreducible highest weight module V (µ) of level k. Proof. We consider the character of V (µ), which is a formal power series defined by (dim V (µ)d,s )q d zs . chq,z V (µ) = q −µ d,s
Since the vectors (4.13) span V (µ), we have chq,z V (µ) ≤ δµ,0 + q −µ
×
1 L1 +L−1 iw(λi−1 ,λi ,λi+1 ) i=1 q (q)L
L≥1 L−1
q
i=1
(λi )
ih(i ,i+1 )
z
L
i=1 i
.
(4.20)
1 ,... ,L =±
Here the second sum is over the sequences (λ0 , . . . , λL ) of integers such that λ0 = µ,
λi+1 = λi ± 1,
0 ≤ λi ≤ k,
λL = 0.
Let us prove the equality in (4.20). Then it implies Theorem 4.6. Set w(λ ˜ i−1 , λi , λi+1 ) := w(λi−1 , λi , λi+1 ) − λi−1 + 2λi − λi+1 . Then we have
" w(λ ˜ i−1 , λi , λi+1 ) =
Hence we find
L−1
q L1 +
i=1
1, λi−1 = λi+1 = λi − 1, 0, otherwise.
iw(λi−1 ,λi ,λi+1 )
= q µ
(λi )
q
L−1 i=1
i w(λ ˜ i−1 ,λi ,λi+1 )
(λi )
=q
µ
(k)
Kµ,(1L ) (q),
(k)
where Kµ,(1L ) (q) is the level-restricted Kostka polynomial. From this formula and 1 ,... ,L =±
q
L−1 i=1
ih(i ,i+1 )
z
L
i=1 i
=
L l=0
(q)L zL−2l , (q)l (q)L−l
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the right-hand side of (4.20) is equal to L≥0
1 (q)L (k) Kµ,(1L ) (q) zL−2l . (q)L (q)l (q)L−l L
(4.21)
l=0
(k)
Here by definition we set Kµ,(10 ) (q) = δµ,0 . Then (4.21) is equal to the character of V (µ) as shown in [2] (the formula (2.14) in the limit N → ∞). Acknowledgements. BF is partially supported by grants RFBR-02-01-01015, RFHR-01-01-00906, INTAS-00-00055. JM is partially supported by the Grant-in-Aid for Scientific Research (B2) no.12440039, and TM is partially supported by (A1) no.13304010, Japan Society for the Promotion of Science. EM is partially supported by the National Science Foundation (NSF) grant DMS-0140460.
References 1. Di Francesco, P., Mathieu, P., S´en´echal, D.: Conformal Field Theory. Springer GTCP, New York: Springer, 1997 2 spaces of coinvariants. 2. Feigin, B., Jimbo, M., Loktev, S., Miwa, T.: Two character formulas for sl Int. J. Mod. Phys. A 19 Suppl.02, 134–154 (2004) 3. Lepowsky, J., Wilson, R.: The structure of standard modules, I: Universal algebras and the Rogers(1) Ramanujan identities. Invent. Math. 77, 199–290 (1984); II: The case A1 , principal gradation, ibid. 79, 417–442 (1985) (1) 4. Lepowsky, J., Primc, M.: Structure of the standard modules for the affine Lie algebra A1 . Contemp. Math. 46, Providence, RI: Am. Math. Soc., 1985, pp. 1–84 (1) 5. Primc, M.: Vertex operator construction of standard modules for An . Pacific J. Math. 162, 143–187 (1994) # C) and combinatorial 6. Meurman, A., Primc, M.: Annihilating fields of standard modules of sl(2, identities. Mem. Am. Math. Soc. 137(652), 1999 7. Andrews, G., Baxter, R., Forrester, P.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys. 35, 193–266 (1984) 8. Forrester, P., Baxter, R.: Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan-type identities. J. Stat. Phys. 38, 435–472 (1985) 9. Huse, D.A.: Exact exponents for infinitely many new multi-critical points. Phys. Rev. B30, 3908–3915 (1984) 10. Feigin, B., Frenkel, E.: Coinvariants of Nilpotent Subalgebras of the Virasoro Algebra and Partition Identities. Adv. Soviet Math. 30, Part I 139–148 (1993) 11. Feigin, B., Jimbo, M., Miwa, T.: Vertex operator algebra arising from the minimal series M(3, p) and monomial basis. In: Proceedings of MathPhys Odessey 2001, Okayama, Basel-Boston: Birkh¨auser, 2003, pp. 179–204 12. Foda, O., Lee, K.S.M., Pugai, Y., Welsh, T.A.: Path generating transforms. q-series from a contemporary perspective (South Hadley, MA, 1998). Contemp. Math. 254, Providence, RI: Amer. Math. Soc., 2000, pp. 157–186 13. Feigin, B., Miwa, T.: Extended vertex operator algebras and monomial bases. In: McGuire Festschrift, Statistical Physics on the Eve of the Twenty-First Century, M. Batchelor et al (eds.), Singapore: World Scientific, 1999 14. Huang, Y.-Z.: Generalized rationality and a “Jacobi identity” for intertwining operator algebras. Selecta Mat. 6, 225–267 (2000) 15. Feigin, B., Nakanishi, T., Ooguri, H.: The annihilating ideals of minimal models. In: Proceedings of the RIMS Research Project 1991, Infinite Analysis B, Advanced Series in Mathematical Physics Vol. 16, Singapore: World Scientific, 1992, pp. 217–238 Communicated by L. Takhtajan
Commun. Math. Phys. 257, 425–471 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1298-5
Communications in
Mathematical Physics
Minimal Dynamical Systems on the Product of the Cantor Set and the Circle Huaxin Lin1, Hiroki Matui2 1 2
Department of Mathematics, East China Normal University, Shanghai, P.R. China Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan. E-mail: [email protected]
Received: 1 June 2004 / Accepted: 9 September 2004 Published online: 25 February 2005 – © Springer-Verlag 2005
Abstract: We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated Cantor minimal system. In the case that cocycles take values in the rotation group, it is also shown that this condition implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately K-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps C(X × T) onto C(X × T). 1. Introduction A celebrated theorem of Giordano, Putnam and Skau [GPS] gave a dynamical characterization of the isomorphism of the crossed product C ∗ -algebras arising from minimal dynamical systems on the Cantor set. The C ∗ -algebra theoretic aspect of this result is indebted to the fact that the algebras are unital simple AT-algebras with real rank zero. From the work of Q. Lin and N. C. Phillips [LP2] and the classification of simple C ∗ algebras of tracial rank zero (see [L4] and [L5]), crossed product C ∗ -algebras arising from minimal diffeomorphisms on a manifold are isomorphic if they have the same Elliott invariants in the case that they have real rank zero (see also [LP1]). However, there are no longer any dynamical characterizations of the isomorphism of these algebras. The notion of approximate conjugacy was introduced in [LM] where it was also suggested that a certain approximate version of conjugacy may be a right equivalence relation to ensure isomorphism of crossed product C ∗ -algebras. Indeed, a complete description was given for Cantor minimal systems. In the present paper, we consider minimal dynamical Current address: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. E-mail: [email protected]
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systems on the product of the Cantor set X and the circle T, and analyze the associated crossed product C ∗ -algebras and approximate conjugacy for those systems. Since the Cantor set X is totally disconnected and the circle T is connected, every minimal dynamical system on X × T can be viewed as a skew product extension of a minimal dynamical system on X. This observation enables us to find a nice “large” subalgebra of the crossed product C ∗ -algebra in a similar fashion to the Cantor case. By applying results of [Ph2], it will be shown that the algebra has stable rank one and satisfies Blackadar’s fundamental comparability property. Moreover, we also prove that the algebra has real rank zero if and only if every invariant measure on X uniquely extends to an invariant measure on X × T. Such a system is said to be rigid. As a special case we consider cocycles taking their values in the rotation group and show that the algebra has tracial rank zero if and only if the system is rigid (see Definition 3.1). By an easy computation of K-groups and the classification theorem of [L5], we conclude that the associated crossed product C ∗ -algebras are actually unital simple AT-algebras of real rank zero. A natural definition of approximate conjugacy for two dynamical systems (X, α) and (Y, β) is the following: there exists a sequence of homeomorphisms σn : X → Y such that lim f ◦ σn ασn−1 − f ◦ β = 0
n→∞
for every f ∈ C(Y ). If such a sequence exists, we can construct an asymptotic homomorphism between the associated crossed product C ∗ -algebras. In [LM], however, it was shown that this simple relation is too weak for Cantor minimal systems. This happens because there is no consistency in the sequence {σn }. Moreover, we proved in [M3] that a similar result holds for minimal dynamical systems on the product of the Cantor set X and the circle T. To obtain a stronger relation, one should impose additional conditions on the conjugating maps σn . We require that σn eventually induces the same map on Ktheory. Approximate K-conjugacy was introduced in [LM] for Cantor minimal systems, and it was shown that two minimal systems are approximately K-conjugate if and only if the associated crossed products are isomorphic. In the present paper, we will show that two minimal systems on X × T associated with cocycles with values in the rotation group are approximately K-conjugate if and only if there is an order and unit preserving isomorphism between the K-theory of two associated crossed products which preserves the images of K-theory of C(X × T). In fact, in the case that both systems are rigid, when two systems are approximately K-conjugate, the associated crossed products are isomorphic. But more is true. They are C ∗ -strongly approximately flip conjugate (see Definition 7.4 below). The only difference from the case of Cantor minimal systems is that we have to control a special projection which does not come from any clopen subsets of X. We call that projection the generalized Rieffel projection. The difference of two generalized Rieffel projections will be written by a Bott element associated with two almost commuting unitaries. A technique developed in [M3, Sect. 4] will be used more carefully in order to fix the position of that projection in the K0 -group. Non-orientation preserving cases are also discussed. By using a Z2 -extension, we can untwist a non-orientation preserving system and obtain an orientation preserving system. The associated crossed product C ∗ -algebra turns out to be isomorphic to the fixed point algebra of a Z2 -action on the crossed product associated with the orientation preserving system. Relation of their K-groups is studied. This paper is organized as follows. In Sect. 2, we collect notations and terminologies relevant to this paper and establish a few elementary facts. In Sect. 3, we investigate
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real rank, stable rank and comparability of projections of the crossed product C ∗ -algebra. Section 4 is devoted to the case that cocycles take values in the rotation group. In Sect. 5, when cocycles take values in the rotation group, we will prove that rigidity implies tracial rank zero. In Sect. 6, the generalized Rieffel projection is defined. In Sect. 7, we show that isomorphism of K-groups implies approximate K-conjugacy when the systems arise from cocycles with values in the rotation group. In Sect. 8, we deal with non-orientation preserving cases. In Sect. 9, we treat examples of various cocycles. 2. Preliminaries Let A be a unital C ∗ -algebra. We denote the union of Mk (A) for k = 1, 2, . . . by M∞ (A). The K0 -group of A is equipped with the order unit [1A ] and the positive cone K0 (A)+ . A homomorphism s : K0 (A) → R is called a state if s carries the order unit to one and the positive cone to nonnegative real numbers. We write the set of all states by S(K0 (A)) and call it the state space. Let T (A) denote the set of all tracial states on A. Endowed with the weak-∗ topology, T (A) is a compact convex set. The space of real valued affine continuous functions on T (A) is written by Aff(T (A)). We denote by D the natural homomorphism from K0 (A) to Aff(T (A)). Namely, D([p])(τ ) is equal to τ (p), where p is a projection of M∞ (A) and τ is a tracial state on A. We say that the algebra A satisfies Blackadar’s second fundamental comparability question when the order on projections of M∞ (A) is determined by traces, that is, if p, q ∈ M∞ (A) are projections and τ (p) < τ (q) for all τ ∈ T (A), then p is Murray-von Neumann equivalent to a subprojection of q. Let X be a compact metrizable space. Equip Homeo(X) with the topology of pointwise convergence in norm on C(X). Thus a sequence {αn }n∈N in Homeo(X) converges to α, if lim sup |f (αn−1 (x)) − f (α −1 (x))| = 0
n→∞ x∈X
for every complex valued continuous function f ∈ C(X). This is equivalent to say that sup d(αn (x), α(x)) x∈X
tends to zero as n → ∞, where d(·, ·) is a metric which induces the topology of X. When X is the Cantor set, this is also equivalent to say that, for any clopen subset U ⊂ X, there exists N ∈ N such that αn (U ) = α(U ) for all n ≥ N . When α : X → X is a homeomorphism on a compact metrizable space X, we denote the crossed product C ∗ -algebra arising from the dynamical system (X, α) by C ∗ (X, α). We use the notation uf u∗ = f ◦ α −1 , where f is a function on X and u is the implementing unitary. If X is infinite and α is minimal, then C ∗ (X, α) is a simple C ∗ -algebra. We regard C(X) as a subalgebra of C ∗ (X, α). But when we need to emphasize the embedding, it will be denoted by jα : C(X) → C ∗ (X, α). Let Mα denote the set of α-invariant probability measures on X. If α is free, then there exists a canonical bijection between Mα and the tracial state space T (C ∗ (X, α)). We may identify these spaces. Let (Y, β) be another dynamical system. A continuous surjection F : X → Y is called a factor map if βF = F α. It is well-known that F yields an affine continuous surjection F∗ : Mα → Mβ by F∗ (µ)(E) = µ(F −1 (E)) for µ ∈ Mα and E ⊂ Y . We also remark that F induces a natural embedding of C ∗ (Y, β) into C ∗ (X, α).
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When α is a minimal homeomorphism on the Cantor set X, we call (X, α) a Cantor minimal system. We briefly review results of [GPS]. The crossed product C ∗ -algebra C ∗ (X, α) is a unital simple AT algebra of real rank zero, and so it can be classified by its K-groups. The K1 -group of C ∗ (X, α) is isomorphic to Z, and K0 (C ∗ (X, α)) is unital order isomorphic to K 0 (X, α) = C(X, Z)/{f − f ◦ α −1 : f ∈ C(X, Z)} equipped with the positive cone K 0 (X, α)+ = {[f ] : f ∈ C(X, Z), f ≥ 0} and the order unit [1X ], where [f ] means its equivalence class. We sometimes write [f ]α to specify α. Throughout this paper, K0 (C ∗ (X, α)) will be identified with K 0 (X, α). By [GPS, Theorem 2.1], K 0 (X, α) is a complete invariant for strong orbit equivalence of Cantor minimal systems. The idea of Kakutani-Rohlin partitions will be used repeatedly. We refer the reader to [HPS, Theorem 4.2] or [M3, Sect. 2] for details. We identify the circle with T = R/Z and write the distance from t ∈ T to zero in T by |t|. Let Rt denote the translation on T = R/Z by t ∈ T. Then {Rt : t ∈ T} forms an abelian subgroup of Homeo(T). We call it the rotation group. The set of isometric homeomorphisms on T is written by Isom(T). Thus, Isom(T) = {Rt : t ∈ T} ∪ {Rt λ : t ∈ T}, where λ ∈ Homeo(T) is defined by λ(t) = −t. The finite cyclic group of order m is denoted by Zm ∼ = Z/mZ and may be identified with {0, 1, . . . , m − 1}. Define o : Homeo(T) → Z2 by 0 ϕ is orientation preserving o(ϕ) = 1 ϕ is orientation reversing. Then the map o(·) is a homomorphism. Let Homeo+ (T) denote the set of orientation preserving homeomorphisms. Note that Homeo+ (T) ∩ Isom(T) consists of rotations. In the rest of this section, we establish notation and some elementary facts concerning dynamical systems on the product of the Cantor set X and the circle T. Lemma 2.1. Let γ be a homeomorphism on X × T. Then, there exist α ∈ Homeo(X) and a continuous map ϕ : X → Homeo(T) such that γ (x, t) = (α(x), ϕx (t)) for all (x, t) ∈ X × T. Proof. This is obvious because the connected component including (x, t) is {x} × T and it must be carried to a connected component by the homeomorphism γ .
We denote the homeomorphism of the form in the lemma above by α × ϕ for short. When ϕx = Rξ(x) with a continuous function ξ : X → T, we write α × Rξ . Let F : X × T → X be the projection onto the first coordinate. Then we have F ◦ (α × ϕ) = α ◦ F . Thus, F is a factor map from (X × T, α × ϕ) to (X, α), and so F induces an affine continuous map from the set of invariant measures of (X × T, α × ϕ) to that of (X, α). Note that if α × ϕ is minimal then α is also minimal. We say that ϕ, ψ : X → Homeo(T) are cohomologous, when there exists a continuous map ω : X → Homeo(T) such that ψx ωx = ωα(x) ϕx for all x ∈ X. If ϕ and ψ are cohomologous, it can be easily verified that α × ϕ and α × ψ are conjugate.
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Let o(ϕ) be the composition of ϕ : X → Homeo(T) and o : Homeo(T) → Z2 , that is, o(ϕ)(x) = o(ϕx ). Under the identification of C(X, Z2 )/{f − f α −1 : f ∈ C(X, Z2 )} with K 0 (X, α)/2K 0 (X, α) (see [M1, Lemma 3.5]), an element of K 0 (X, α)/2K 0 (X, α) is obtained from o(ϕ). We write it by [o(ϕ)] or [o(ϕ)]α . Definition 2.2. Let (X, α) be a Cantor minimal system and ϕ : X → Homeo(T) be a continuous map. We say that α × ϕ or ϕ is orientation preserving, if [o(ϕ)] is zero in K 0 (X, α)/2K 0 (X, α). Notice that the concept of ‘orientation reversing’ does not make sense in this situation and that (α × ϕ)2 may not be orientation preserving. Lemma 2.3 ([M3, Lemma 4.3]). Let (X, α) be a Cantor minimal system. If α × ϕ is an orientation preserving homeomorphism on X × T, then there exists a continuous map ψ : X → Homeo+ (T) such that ϕ is cohomologous to ψ. Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuous map. We would like to compute the K-groups of C ∗ (X ×T, α ×ϕ). Let us begin with the orientation preserving case. When [o(ϕ)] is zero in K 0 (X, α)/2K 0 (X, α), by Lemma 2.3, we may assume that o(ϕ)(x) is zero for all x ∈ X. It is evident that α × ϕ induces the action α ∗ : f → f ◦ α −1 on K0 (C(X × T)) ∼ = C(X, Z), and that the kernel of id −α ∗ is spanned by [1X ] and the ∗ cokernel of id −α is K 0 (X, α). Since o(ϕ)(x) = 0 for all x ∈ X, the induced action on K1 (C(X × T)) ∼ = C(X, Z) is α ∗ , too. Consequently we have the following. Lemma 2.4. Let (X, α) be a Cantor minimal system and let α × ϕ be an orientation preserving homeomorphism on X × T. Then both K0 (C ∗ (X × T, α × ϕ)) and K1 (C ∗ (X × T, α × ϕ)) are isomorphic to Z ⊕ K 0 (X, α). Of course the embedding C ∗ (X, α) ⊂ C ∗ (X × T, α × ϕ) induces the embedding of into K0 (C ∗ (X × T, α × ϕ)) preserving the unit. When α × ϕ is minimal, it can be easily verified that this is really an order embedding. The order structure of the whole K0 -group will be apparent in later sections. Next, let us consider the non-orientation preserving case. Suppose that [o(ϕ)] is not zero in K 0 (X, α)/2K 0 (X, α). Clearly α×ϕ induces the same action on K0 (C(X×T)) ∼ = C(X, Z) as the orientation preserving case. But, on the K1 -group, the induced action is different. It is given by K 0 (X, α)
αϕ∗ (f )(x) = (−1)o(ϕ)(α
−1 (x))
f (α −1 (x))
for f ∈ C(X, Z). We need to know the kernel and the cokernel of id −αϕ∗ . Suppose that f ∈ C(X, Z) belongs to Ker(id −αϕ∗ ) and f = 0. Since |f (x)| = |αϕ∗ (f )(x)| = |α ∗ (f )(x)| for all x ∈ X, the minimality of α implies that |f (x)| is a constant function. −1 Define c ∈ C(X, Z2 ) by f (x) = (−1)c(x) |f (x)|. Then f (x)= (−1)o(ϕ)(α (x)) f (α −1 (x)) yields c(x) = o(ϕ)(α −1 (x)) + c(α −1 (x)), which contradicts [o(ϕ)] = 0. Hence id −αϕ∗ is injective. It follows that K0 (C ∗ (X × T, α × ϕ)) ∼ = K 0 (X, α).
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Let (X × Z2 , α × o(ϕ)) be the skew product extension associated with the Z2 -valued cocycle o(ϕ). As [o(ϕ)] = 0, (X × Z2 , α × o(ϕ)) is a Cantor minimal system (see [M1, Lemma 3.6]). By definition, K 0 (X × Z2 , α × o(ϕ)) is isomorphic to the cokernel of id −(α × o(ϕ))∗ on C(X × Z2 , Z). Let π be the projection from X × Z2 onto the first coordinate. Then π is a factor map from (X × Z2 , α × o(ϕ)) to (X, α). It is well-known that [f ] → [f ◦ π] is an order embedding between the K 0 -groups, and so we will regard K 0 (X, α) as a subgroup of K 0 (X × Z2 , α × o(ϕ)). It is convenient to introduce a monomorphism δ from C(X, Z) to C(X × Z2 , Z) defined by δ(f )(x, k) = (−1)k f (x). Then we can check that δ ◦ αϕ∗ = (α × o(ϕ))∗ ◦ δ. Define γ ∈ Homeo(X × Z2 ) by γ (x, k) = (x, k + 1). Note that γ commutes with α × o(ϕ). Since Im δ = Im(id −γ ∗ ), we have Coker(id −αϕ∗ ) ∼ = Im δ/ Im δ ◦ (id −αϕ∗ ) = Im(id −γ ∗ )/ Im((id −(α × o(ϕ))∗ ) ◦ (id −γ ∗ )) ∼ = C(X × Z2 , Z)/(Ker(id −γ ∗ ) + Im(id −(α × o(ϕ))∗ )) ∼ = K 0 (X × Z2 , α × o(ϕ))/K 0 (X, α). We can summarize the conclusion just obtained as follows. Lemma 2.5. Let (X, α) be a Cantor minimal system and let α × ϕ be a non-orientation preserving homeomorphism on X × T. Then we have K0 (C ∗ (X × T, α × ϕ)) ∼ = K 0 (X, α) and K1 (C ∗ (X × T, α × ϕ)) ∼ = Z ⊕ K 0 (X × Z2 , α × o(ϕ))/K 0 (X, α), where the Z-summand of the K1 -group is generated by the implementing unitary. By the same reason as orientation preserving cases, one sees that K0 (C ∗ (X × T, α × ϕ)) is unital order isomorphic to K 0 (X, α), when α × ϕ is minimal. We remark that the torsion subgroup of K 0 (X × Z2 , α × o(ϕ))/K 0 (X, α) is isomorphic to Z2 by [M3, Lemma 4.5]. The torsion element is given by f0 (x, k) =
1 0
o(ϕ)(α −1 (x)) = 1 and k = 0 otherwise.
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3. Real Rank and Stable Rank Let (X, α) be a Cantor minimal system and let X x → ϕx ∈ Homeo(T) be a continuous map. In this section, we would like to compute the real rank and the stable rank of C ∗ (X × T, α × ϕ). Let us denote the crossed product C ∗ -algebra by A and the implementing unitary by u. Our crucial tool is a “large” subalgebra of A, which is described below. The proof will be done by applying the argument of [Ph2] to that subalgebra. We would like to begin with the definition of the rigidity. Definition 3.1. Let α × ϕ be a homeomorphism on the product of the Cantor set X and the circle T. We say that α × ϕ is rigid, if the canonical factor map from (X × T, α × ϕ) to (X, α) induces an isomorphism between the sets of invariant probability measures. Remark 3.2. In the definition above, even if α is minimal and α × ϕ is rigid, α × ϕ may not be minimal. For example, let (X, α) be an odometer system and let ϕ ∈ Homeo+ (T) be a Denjoy homeomorphism, that is, the rotation number of ϕ is irrational but ϕ is not conjugate to a rotation. It is well-known that ϕ has a unique invariant nontrivial closed subset Y , and so α × ϕ has a nontrivial closed invariant subset X × Y . Thus α × ϕ is not minimal. It is also known that Y is the Cantor set and ϕ|Y is minimal. This Cantor minimal system is called a Denjoy system (see [PSS] for details). The complement of Y consists of countable disjoint open intervals and each interval is a wandering set. Hence, for every α × ϕ-invariant probability measure µ, we have µ(X × Y c ) = 0. Moreover, as pointed out in [M1, Sect. 7 (2)], the product of (X, α) and (Y, ϕ|Y ) is uniquely ergodic. It follows that α × ϕ is uniquely ergodic. In particular, it is rigid. For x ∈ X, let Ax be the C ∗ -subalgebra generated by C(X × T) and uC0 ((X \ {x}) × T). In [Pu1, Theorem 3.3], it was proved that Ax ∩C ∗ (X, α) is a unital AF algebra, where we regard C ∗ (X, α) as a C ∗ -subalgebra of A. This AF subalgebra played a crucial role in the subsequent papers [HPS] and [GPS]. In our situation, the C ∗ -subalgebra Ax is not an AF algebra but an AT algebra, and it helps us to show real rank zero. Proposition 3.3. In the setting above, we have the following. (1) Ax is a unital AT algebra. (2) When α×ϕ is orientation preserving, K0 (Ax ) is unital order isomorphic to K 0 (X, α) and K1 (Ax ) is isomorphic to K 0 (X, α). (3) When α × ϕ is not orientation preserving, K0 (Ax ) is unital order isomorphic to K 0 (X, α) and K1 (Ax ) is isomorphic to an extension of Coker(id −αϕ∗ ) by Z. (4) There exists a canonical bijection between the tracial state space T (Ax ) and the set of α × ϕ-invariant probability measures. (5) Ax is simple if and only if α × ϕ is minimal. (6) If α × ϕ is minimal, then Ax is real rank zero if and only if α × ϕ is rigid. Proof. (1) Let Pn = {X(n, v, k) : v ∈ Vn , k = 1, 2, . . . , h(v)} be a sequence of Kakutani-Rohlin partitions which gives a Bratteli-Vershik model for (X, α) (see [HPS, Theorem 4.2] or [M3, Sect. 2] for Kakutani-Rohlin partitions). We assume that the sequence of the roof sets R(Pn ) = X(n, v, h(v)) v∈V
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shrinks to a singleton {x}. Let An be the C ∗ -subalgebra generated by C(X × T) and uC(R(Pn )c × T). It is easy to see that Ax is the norm closure of the union of all An ’s. By using a similar argument to [Pu1, Lemma 3.1], it can be shown that An is isomorphic to Mh(v) ⊗ C(X(n, v, h(v))) ⊗ C(T), v∈Vn
which is an AT algebra. To verify this, define a projection pv by pv =
h(v)
1X(n,v,k)
k=1
for each v ∈ V . Since pv u(1 − 1R(Pn ) ) = u(1 − 1R(Pn ) )pv , the projection pv is central in An . Clearly ui−j 1X(n,v,j ) for i, j = 1, 2, . . . , h(v) form matrix units of pv An pv . By 1X(n,v,h(v)) An 1X(n,v,h(v)) = C(X(n, v, h(v)) × T), we obtain the description above. Hence Ax is also an AT algebra. (2) There is a natural homomorphism from Ki (C(X × T)) ∼ = C(X, Z) to Ki (An ) for i = 1, 2. By (1), the kernel of this map is {f − f ◦ α −1 : f ∈ C(X, Z), f (y) = 0 for all y ∈ R(Pn )}. Therefore Ki (Ax ) is isomorphic to C(X, Z)/{f − f ◦ α −1 : f ∈ C(X, Z), f (x) = 0}. It follows from 1X ◦ α −1 = 1X that {f − f ◦ α −1 : f ∈ C(X, Z), f (x) = 0} = {f − f ◦ α −1 : f ∈ C(X, Z)}, which implies Ki (Ax ) ∼ = K 0 (X, α). (3) The computation of the K0 -group is the same as the orientation preserving case. Let us consider K1 (Ax ). It is not hard to see that K1 (Ax ) is isomorphic to C(X, Z)/{f − αϕ∗ (f ) : f ∈ C(X, Z) and f (x) = 0}. We follow the notation used in the discussion before Lemma 2.5. The image of {f − αϕ∗ (f ) : f ∈ C(X, Z) and f (x) = 0} by δ is equal to the image of {f − f ◦ (α × o(ϕ))−1 : f ∈ C(X × Z2 , Z), f (x, 0) = (x, 1) = 0} by id −γ ∗ . Hence, in the same way as Lemma 2.5, we have K1 (Ax ) ∼ = K 0 (X × Z2 , α × o(ϕ); Z2 )/K 0 (X, α). See [M2] for the definition of K 0 (X × Z2 , α × o(ϕ); Z2 ). By [Pu1, Theorem 4.1], 0 → Z → K 0 (X × Z2 , α × o(ϕ); Z2 ) → K 0 (X × Z2 , α × o(ϕ)) → 0
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is exact, which implies that K1 (Ax ) is an extension of Coker(id −αϕ∗ ) ∼ = K 0 (X × Z2 , α × o(ϕ))/K 0 (X, α) by the integers Z. (4) Since there exists a canonical bijection between T (A) and the set of α × ϕinvariant probability measures, it suffices to check that every τ ∈ T (Ax ) extends to a tracial state on A. Let f ∈ C(X × T) be a function satisfying 0 ≤ f ≤ 1. Take a natural number n ∈ N arbitrarily. We can find a clopen neighborhood U of x such that U, α −1 (U ), . . . , α −n (U ) are mutually disjoint. Let p = 1U ×T ∈ C(X × T). Then we have τ (p) < n−1 , because p, u∗ pu, . . . , un∗ pun are mutually equivalent in Ax and mutually disjoint. We also notice that u(1 − p)f belongs to Ax . It follows that τ (f ) ≤ τ (p) + τ ((1 − p)f ) = τ (p) + τ (u(1 − p)f u∗ ) <
1 + τ (uf u∗ ). n
Similarly we can see τ (uf u∗ ) < n−1 + τ (f ). Since n is arbitrary, τ (f ) equals τ (uf u∗ ), which means that τ extends to a trace on A. (5) Note that the C ∗ -algebra A can be regarded as a groupoid C ∗ -algebra associated with the equivalence relation R = {(z, (α × ϕ)k (z)) : z ∈ X × T, k ∈ Z}. Then the C ∗ -subalgebra Ax corresponds to the subequivalence relation Rx = R \ {((α × ϕ)k (x, t), (α × ϕ)l (x, t)) : t ∈ T, (1 − k, l) ∈ N2 or (k, 1 − l) ∈ N2 }. It is well-known that a homeomorphism on a compact space is minimal if and only if every positive orbit is dense. Therefore α × ϕ is minimal if and only if each equivalence class of Rx is dense in X × T. It follows from [R, Proposition 4.6] that this is equivalent to Ax being simple. (6) By (1) and (5), Ax is a unital simple AT algebra. By (2) and (3), in the K0 -group, every projection of Ax is equivalent to some [f ] ∈ K 0 (X, α). Hence, projections in Ax separate traces on Ax if and only if α × ϕ is rigid. Then the conclusion follows from [BBEK, Theorem 1.3].
Remark 3.4. The six-term exact sequence of [Pu2, Theorem 2.4] applies to this situation. See [Pu2, Example 2.6]. The reduced groupoid C ∗ -algebra Cr∗ (H ) appearing there is isomorphic to C(T) ⊗ K. We would like to consider the real rank of A. In [Ph2], it was shown that if G is an almost AF Cantor groupoid and Cr∗ (G) is simple, then Cr∗ (G) has real rank zero. The key of its proof was the presence of a “large” AF subalgebra. We will show that a similar argument is possible when the AF subalgebra is replaced by a subalgebra with tracial rank zero. Definition 3.5 ([L5, Theorem 6.13]). We say that a unital simple C ∗ -algebra A has tracial (topological) rank zero, if for any finite subset F ⊂ A, any ε > 0 and any nonzero positive element c ∈ A, there exists a projection e ∈ A and a finite dimensional unital subalgebra E ⊂ eAe (that is, e is the identity of E) such that: (1) ae − ea < ε for all a ∈ F. (2) For every a ∈ F, there is b ∈ E such that pap − b < ε. (3) 1 − e is Murray-von Neumann equivalent to a projection in cAc.
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Definition 3.6. A unital simple C ∗ -algebra A is called an almost tracially AF algebra, if there exists a unital simple subalgebra B ⊂ A with tracial rank zero such that the following holds: for any finite subset F ⊂ A and any ε > 0, there exists a projection p ∈ B such that: (1) For every a ∈ F, there is b ∈ B such that ap − b < ε. (2) τ (1 − p) < ε for every tracial state τ ∈ T (B). Lemma 3.7. Suppose that α × ϕ is a minimal homeomorphism on X × T. If α × ϕ is rigid, then the crossed product C ∗ -algebra A = C ∗ (X × T, α × ϕ) is an almost tracially AF algebra. Proof. Take x ∈ X. By Proposition 3.3, Ax is a unital simple AT algebra of real rank zero. It follows from [L1, Proposition 2.6] that Ax is a unital simple algebra with tracial rank zero. Suppose that a finite subset F ⊂ A and ε > 0 are given. Since N k fk u : N ∈ N, fk ∈ C(X × T) k=−N
is a dense subalgebra of A, we may assume that there is N ∈ N such that F is contained in N k EN = fk u : fk ∈ C(X × T) . k=−N
We can find a clopen neighborhood U of x so that α 1−N (U ), α 2−N (U ), . . . , U, α(U ), . . . , α N (U ) are mutually disjoint and µ(U ) < ε/2N for all µ ∈ Mα . Put V = N k k 1−k p k=1−N α (U ) and p = 1V c ×T ∈ C(X × T). It is easy to see that u p and u belong to Ax for k = 0, 1, . . . , N, and so ap ∈ Ax for every a ∈ EN . Moreover, τ (1 − p) is less than ε for every τ ∈ T (Ax ). This finishes the proof of the lemma.
The following lemma is a generalization of [Ph2, Lemma 4.3]. Lemma 3.8. Let A be a unital simple algebra with tracial rank zero. Let p ∈ A be a projection and let a ∈ A be a nonzero self-adjoint element. For any ε > 0 and n ∈ N satisfying nε > 1, there exists a projection q ∈ A such that qa − aq < εa, p ≤ q and τ (q) < (2n + 1)τ (p) for all τ ∈ T (A). Proof. Without loss of generality, we may assume a = 1. Put ε0 = 8−1 (ε − n−1 ). Choose δ0 > 0 so that whenever p, q ∈ A are projections satisfying pq − p < δ0 , then there exists a unitary u ∈ A such that u − 1 < ε0 and p ≤ uqu∗ . Let δ = min{ε0 , 4−1 δ0 , (2n + 1)−1 τ (p) : τ ∈ T (A)}. By [L1, Proposition 2.4] together with results of [L1, Section 3], there exist a projection e ∈ A and a finite dimensional unital subalgebra E ⊂ eAe such that:
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• ae − ea < δ and pe − ep < δ. • There exist b, c ∈ E such that eae − b < δ and epe − c < δ. • τ (1 − e) < δ for every tracial state τ ∈ T (A). We may assume that b = b∗ , b = 1 and c is a projection. Thanks to [Ph2, Lemma 4.2], there exists a projection q0 ∈ E such that c ≤ q0 , [q0 ] ≤ 2n[c] ∈ K0 (E), and q0 b − bq0 <
1 . n
Put q = 1 − e + q0 . Since pq − p ≤ (p(1 − e) + peq0 ) − (p(1 − e) + pcq0 ) +(p(1 − e) + pcq0 ) − (p(1 − e) + pe) < 2δ + 2δ ≤ δ0 , there is a unitary u ∈ A such that u − 1 < ε0 and p ≤ uqu∗ . It is not hard to see that τ (uqu∗ ) = τ (q) = τ (1 − e) + τ (q0 ) < δ + 2nτ (c) < (2n + 1)δ + 2nτ (pe) < (2n + 1)τ (p) for all τ ∈ T (A) and that [uqu∗ , a] < [q, a] + 4ε0 < [q, b] + 4δ + 4ε0 1 < + 8ε0 = ε, n thereby completing the proof.
The following is a well-known matrix trick. We omit the proof. Lemma 3.9. When a is a self-adjoint element of a unital C ∗ -algebra A, a0 00 is approximated by an invertible self-adjoint element in A ⊗ M2 . Although the proof of the following theorem is almost the same as that of [Ph2, Theorem 4.7], we would like to state it for the reader’s convenience. Theorem 3.10. If a unital simple C ∗ -algebra A is an almost tracially AF algebra, then A has real rank zero. Proof. Let B ⊂ A be a unital simple subalgebra with tracial rank zero as in Definition 3.6. Let a ∈ A be self-adjoint and non-invertible. It suffices to show that a is approximated by a self-adjoint invertible element of A. Without loss of generality, we may assume a ≤ 1. Take ε > 0 arbitrarily. Define a continuous function g on [−1, 1] by 1 − ε−1 |t| |t| ≤ ε g(t) = 0 otherwise.
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Put ε0 = min{τ (g(a)) : τ ∈ T (A)}. Since A is simple, ε0 is positive. Applying [Ph2, Lemma 4.4] to g : [−1, 1] → [0, 1] and 4−1 ε0 > 0, we obtain δ > 0. We may assume that δ is less than ε. Choose a natural number n ∈ N so that
ε0 δ 1 < min , . n 12 2 By definition, there is a projection p ∈ B such that a(1 − p) is close to B and τ (p) is less than n−2 for all τ ∈ T (B). By perturbing a, we may assume that a(1 − p) belongs to B. Then, we can apply Lemma 3.8 to a − pap ∈ B and p ∈ B, and get a projection q ∈ B such that p ≤ q, [q, a − pap] <
δ a − pap ≤ δ 2
and τ (q) ≤ (2n + 1)τ (p) <
ε0 3 < n 4
for every tracial state τ ∈ T (B). It follows that ε0 ε0 > . 4 4 We also notice that [q, a] < δ. Put a0 = (1 − q)a(1 − q) ∈ B. By the choice of δ, we have ε0 ε0 > . τ (g(a0 )) > τ (g(a)) − τ (q) − 4 2 A unital simple algebra with tracial rank zero has real rank zero by [L1, Theorem 3.4], and so there exists a projection r in the hereditary subalgebra of B generated by g(a0 ) such that ε0 rg(a0 ) − g(a0 ) < . 4 Then ε0 ε0 > τ (r) ≥ τ (rg(a0 )r) > τ (g(a0 )) − 4 4 for all τ ∈ T (B). Since the order on projections of B is determined by traces (see [L2, Theorem 6.8, 6.13]), there is a projection r0 ∈ B such that r0 ≤ r and r0 ∼ q. Moreover, by means of [Ph2, Lemma 4.6], we have τ (g(a)) − τ (q) > ε0 −
r0 a0 − a0 r0 < 2ε and r0 a0 r0 < ε. As a result, 2ε
4ε
a ≈ qaq + a0 ≈ qaq + r0 a0 r0 + (1 − q − r0 )a0 (1 − q − r0 ) ε
≈ qaq + (1 − q − r0 )a0 (1 − q − r0 ) is obtained. The element (1 − q − r0 )a0 (1 − q − r0 ) belongs to B and B has real rank zero. By applying Lemma 3.9 to qaq and r0 ∼ q, we can get the conclusion.
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Corollary 3.11. For a minimal homeomorphism α × ϕ on X × T, the following are equivalent. (1) α × ϕ is rigid. (2) The crossed product C ∗ -algebra A = C ∗ (X × T, α × ϕ) has real rank zero. (3) D(K0 (A)) is uniformly dense in Aff(T (A)). Proof. (1)⇒(2). This is immediate from Lemma 3.7 and the theorem above. (2)⇒(3). Since α × ϕ is minimal, A is simple. By Theorem 3.12, A is stably finite and the projections in A ⊗ K satisfy cancellation. Furthermore, K0 (A) is weakly unperforated by [Ph1, Theorem 4.5]. It follows from [B, Theorem 6.9.3] that the image of K0 (A) is uniformly dense in real valued affine continuous functions on QT (A), the set of quasitraces on A. Because every element of Aff(T (A)) comes from a self-adjoint element of A (see [BKR, Proposition 3.12] for instance), it extends to a real valued affine continuous function on QT (A). Hence D(K0 (A)) is uniformly dense in Aff(T (A)). Note that if one uses the deep result obtained by Haagerup in [H], the latter half of the proof is superfluous. (3)⇒(1). Suppose that α × ϕ is not rigid. Thus, there are ν1 , ν2 ∈ Mα×ϕ such that F∗ (ν1 ) = F∗ (ν2 ) = ν ∈ Mα , where F is the canonical factor map onto (X, α). Let τ1 and τ2 be the tracial states on A arising from ν1 and ν2 . Since τ1∗ ([f ]) = ν1 (f ◦ F ) = ν(f ) = ν2 (f ◦ F ) = τ2∗ ([f ]) for all [f ] ∈ K 0 (X, α), projections in C ∗ (X, α) cannot separate traces on A. Therefore, we can finish the proof here when α × ϕ is not orientation preserving. Assume that α × ϕ is orientation preserving. By Lemma 2.4, K0 (A) is isomorphic to Z ⊕ K 0 (X, α). Suppose that there are projections e1 , e2 ∈ Mk (A) for some integer k such that [e1 ] − [e2 ] = (1, 0) ∈ Z ⊕ K 0 (X, α). Then, for any x = (n, [f ]) ∈ K0 (A) ∼ = Z ⊕ K 0 (X, α), we have τ1∗ (x) − τ2∗ (x) = n(τ1 (e1 − e2 ) − τ2 (e1 − e2 )). Hence {τ1∗ (x) − τ2∗ (x) : x ∈ K0 (A)} is discrete in R, which is a contradiction.
We now turn to a consideration of stable rank of A. Suppose that α × ϕ is minimal but may not be rigid. The C ∗ -subalgebra Ax may not have real rank zero. But, it is still a unital simple AT algebra by Proposition 3.3. Moreover, Lemma 3.7 is also valid when one replaces ‘almost tracially AF’ by ‘almost AT’. A unital simple AT algebra is known to have property (SP), that is, every nonzero hereditary subalgebra contains a nonzero projection. Hence we see that A also has property (SP) by virtue of Lemma 3.7. We also remark that Ax has stable rank one and the order on projections of Ax is determined by traces, because Ax is a unital simple AT algebra. Then, by reading the proof of [Ph2, Theorem 5.2] carefully, it turns out that A and the “large” subalgebra Ax do not need to have real rank zero and that they only need to have property (SP) so that the proof works. As a consequence, we have the following. Theorem 3.12. When α ×ϕ is a minimal homeomorphism on X ×T, the crossed product C ∗ -algebra A = C ∗ (X × T, α × ϕ) has stable rank one. In particular, the projections in A ⊗ K satisfy cancellation.
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In [Ph1, Theorem 4.5], it was proved that A satisfies the K-theoretic version of Blackadar’s second fundamental comparability question, that is, if x ∈ K0 (A) satisfies τ∗ (x) > 0 for all τ ∈ T (A), then x ∈ K0 (A)+ . In particular, K0 (A) is weakly unperforated. Combining this with the theorem above, we can deduce the following. Theorem 3.13. When α ×ϕ is a minimal homeomorphism on X ×T, the order on projections of M∞ (C ∗ (X×T, α×ϕ)) is determined by traces. In other words, C ∗ (X×T, α×ϕ) satisfies Blackadar’s second fundamental comparability question. 4. Cocycles with Values in the Rotation Group Let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map. In this section, we would like to investigate a homeomorphism α × Rξ on X × T and its related crossed product C ∗ -algebra A = C ∗ (X × T, α × Rξ ). Of course, α × Rξ is orientation preserving. Definition 4.1. Let α be a minimal homeomorphism on X. Define KT0 (X, α) = C(X, T)/{η − ηα −1 : η ∈ C(X, T)}. The equivalence class of ξ ∈ C(X, T) in KT0 (X, α) is denoted by [ξ ]α or [ξ ]. Let θ ∈ T and put ξ(x) = θ for all x ∈ X. Thus, ξ is a constant function. It is easy to see that [ξ ] is zero in KT0 (X, α) if and only if θ is a topological eigenvalue of (X, α). The reader may refer to [W, Theorem 5.17] for topological eigenvalues. Since X is compact, the set of topological eigenvalues is at most countable. It follows that KT0 (X, α) is uncountable. At first, we describe when α × Rξ is minimal in terms of KT0 (X, α). Note that more general results were obtained in [Pa]. Lemma 4.2. Let (X, α) be a Cantor minimal system and ξ : X → T be a continuous map. Then, α × Rξ is minimal if and only if n[ξ ] = 0 in KT0 (X, α) for all n ∈ N. Proof. Suppose that there exist n ∈ N and η such that nξ = η − ηα −1 . Then {(x, t) ∈ X × T : nt = η(α −1 (x))} is closed and invariant under α × Rξ , and so α × Rξ is not minimal. Let us prove the other implication. Assume that α × Rξ is not minimal. Let E be a minimal subset of α × Rξ . Note that id ×Rt commutes with α × Rξ . Since E is not the whole of X × T, G = {t ∈ T : (id ×Rt )(E) = E} is a closed proper subgroup of T. It follows that there exists n ∈ N such that G = {t ∈ T : nt = 0}. Moreover, on account of the minimality of E, we can deduce that there exists η : X → T such that E = {(x, t) ∈ X × T : nt = η(x)}. The map η is continuous, because E is closed. Hence we have nξ = ηα − η.
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If α × Rξ is not minimal, then there exist uncountably many minimal closed subsets. In particular, it is not rigid. Compare this with Remark 3.2. Lemma 4.3. Let (X, α) and (Y, β) be Cantor minimal systems. Let ξ and ζ be continuous maps from X to T. Suppose that α × Rξ is minimal. Then, α × ξ and β × ζ is conjugate if and only if there exists a homeomorphism F : X → Y such that F α = βF , and [ξ ]α = [ζ F ]α or [ξ ]α = −[ζ F ]α in KT0 (X, α). Proof. The ‘if’ part is clear. We consider the ‘only if’ part. Let F × ϕ : X × T → Y × T be a conjugating map, that is, F α = βF and ϕα(x) (s + ξ(x)) = ϕx (s) + ζ (F (x)) for all (x, s) ∈ X × T. For every t ∈ T, id ×Rt commutes with β × Rζ , and so (F ×ϕ)−1 (id ×Rt )(F ×ϕ) commutes with α×Rξ . Let x ∈ X and put s = ϕx−1 (ϕx (0)+t). Then we have (F × ϕ)−1 (id ×Rt )(F × ϕ)(x, 0) = (x, s) = (id ×Rs )(x, 0). By the minimality of α × ξ , we can conclude that (F × ϕ)−1 (id ×Rt )(F × ϕ) = id ×Rs . It follows that the mapping t → s is a continuous injective homomorphism from T to T. Thus, (F × ϕ)−1 (id ×Rt )(F × ϕ) = id ×Rt for all t ∈ T, or (F × ϕ)−1 (id ×Rt )(F × ϕ) = id ×R−t for all t ∈ T. Without loss of generality we may assume the first, which yields ϕx (s + t) = ϕx (s) + t for all (x, s) ∈ X × T and t ∈ T. It follows that ϕx equals Rϕx (0) and ξ(x) + ϕα(x) (0) = ϕx (0) + ζ (F (x)). Thereby the assertion follows.
Next, we would like to consider when α × Rξ is rigid. Although the following is a special case of [Pa, Theorem 3] or [Z, Theorem 3.5], we include the proof for the reader’s convenience. Lemma 4.4. Let (X, α) be a Cantor minimal system. For a continuous function ξ : X → T, the following are equivalent. (1) α × Rξ is rigid. (2) Every α × Rξ -invariant measure ν is also id ×Rt -invariant for all t ∈ T, that is, ν is a product measure of the Haar measure on T and an invariant measure for (X, α). (3) For every α-invariant measure µ on X and n ∈ N, there does not exist a Borel function η : X → T such that nξ(x) = η(x) − ηα −1 (x) for µ-almost every x ∈ X.
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Proof. We denote the canonical factor map from (X × T, α × Rξ ) to (X, α) by π . (1)⇒(2). This is immediate from π∗ ◦ (id ×Rt )∗ = (π ◦ (id ×Rt ))∗ = π∗ . (2)⇒(1). Define a continuous map : C(X × T) → C(X) by (f )(x) = f (x, t) dt. T
If ν ∈ Mα×Rξ is id ×Rt -invariant, then ν(f ) = π∗ (ν)((f )) for all f ∈ C(X × T). Hence π∗−1 (π∗ (ν)) = {ν}. (2)⇒(3). Suppose that there exist an α-invariant measure µ ∈ Mα , n ∈ N and a Borel function η : X → T such that nξ(x) = η(x) − ηα −1 (x) for µ-almost every x ∈ X. Then C(X × T) f →
1 n
X
f (x, t) dµ(x) ∈ C
nt=ηα −1 (x)
yields a probability measure on X × T. Note that the summation runs over n distinct t’s which satisfy nt = ηα −1 (x). This measure is α × Rξ -invariant, because we have f (α(x), t + ξ(x)) = f (α(x), t) nt=ηα −1 (x)
nt=ηα −1 (α(x))
for µ-almost every x ∈ X. But it is not the product of the Haar measure and µ. (3)⇒(2). Suppose that ν ∈ Mα×Rξ is not invariant under the rotation id ×Rt . We may assume that ν is an ergodic measure. It follows from [KH, Corollary 4.1.9] or [W, Lemma 6.13] that there exists an Fσ subset E ⊂ X × T such that ν(E) = 1 and 1 f ((α × Rξ )k (x, s)) = ν(f ) n→∞ n n−1
lim
k=0
for all f ∈ C(X × T) and (x, s) ∈ E. We may assume that E is α × Rξ -invariant. Put G = {t ∈ T : (id ×Rt )∗ (ν) = ν}. By assumption, G is a closed proper subgroup of T. Thus, there is n ∈ N such that G = {t ∈ T : nt = 0}. If (x, s) belongs to E, then we have 1 f ((α × Rξ )k (x, s + t)) = ν(f ◦ (id ×Rt )) = (id ×Rt )∗ (ν)(f ) n→∞ n n−1
lim
k=0
/ G. Furthermore, by for all f ∈ C(X × T). Therefore (id ×Rt )(E) ∩ E is empty if t ∈ replacing E by (id ×Rt )(E), t∈G
we may assume that (id ×Rt )(E) = E for all t ∈ G. On the Fσ subset π(E) ⊂ X, we define a T-valued function η by η(x) = nt for (x, t) ∈ E. If E0 ⊂ E is closed, then η
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is evidently continuous on π(E0 ). It follows that η is a well-defined Borel function. For (x, s) ∈ E, (α × Rξ )(x, s) = (α(x), s + ξ(x)) belongs to E, and so η(α(x)) = ns + nξ(x) = α(x) + nξ(x) is obtained. Since this equation holds for all x ∈ π(E) and π∗ (ν)(π(E)) ≥ ν(E) = 1, the proof is completed.
Let µ ∈ Mα . As in the discussion following Definition 4.1, let ξ(x) = θ be a constant function. Then, there exists a Borel function η ∈ C(X, T) such that ξ(x) = η(x) − ηα −1 (x) √
for µ-almost every x ∈ X if and only if e2π −1θ is an eigenvalue of the unitary operator πµ (uα ) ∈ L2 (X, µ), where πµ is a representation of C ∗ (X, α) corresponding to the invariant measure µ. Since L2 (X, µ) is separable, eigenvalues of a unitary operator are at most countable. Hence, by the lemma above, we can obtain a lot of rigid homeomorphisms. Moreover, it is known that eigenvalues of the unitary operator πµ (uα ) need not be topological eigenvalues of (X, α). Therefore we can see that there exists a minimal homeomorphism α × Rξ which is not rigid. We will look at a concrete example in Example 9.1. There is another way to find a rigid homeomorphism. Let ξ ∈ C(X, T) and let ˜ξ ∈ C(X, R) be its lift (this is always possible because X is the Cantor set). Then µ → µ(ξ˜ ) gives an affine function from the set of α-invariant measures Mα to R. By the lemma above, if nµ(ξ˜ ) ∈ / µ(C(X, Z)) for each ergodic α-invariant measure µ and n ∈ N, then α × Rξ is rigid. Next, we will show that A = C ∗ (X×T, α×Rξ ) can be written as a crossed product of ∗ ι(ξ ) on C ∗ (X, α) by ι(ξ )(f ) = f C (X, α) by a certain action. Define an automorphism √ for all f ∈ C(X) and ι(ξ )(uα ) = uα e2π −1ξ(x) , where uα denotes the implementing unitary of C ∗ (X, α). This kind of automorphism was considered in [M1]. We remark that ι(ξ ) is approximately inner, because ι(ξ )∗ is the identity on the K-groups (or one can deduce it from Lemma 6.1 or [M1, Lemma 5.1]). Let ιˆ(ξ ) denote the dual action on C ∗ (X, α) ι(ξ ) Z. Proposition 4.5. There is an isomorphism π from the crossed product C ∗ -algebra A = C ∗ (X × T, α × Rξ ) to C ∗ (X, α) ι(ξ ) Z such that π(f ) = f for all f ∈ C(X) and π(g ◦ (id ×Rt )) = ιˆ(ξ )t (π(g)) for all g ∈ C(X × T) and t ∈ T. Proof. In order to avoid confusion, we have to use different symbols for three implementing unitaries: we denote the implementing unitary in C ∗ (X, α) by uα and denote the unitary implementing ι(ξ ) by v, while u ∈ A denotes the unitary√ implementing α × Rξ . Let z ∈ C(X × T) be a unitary defined by z(x, t) = e2π −1t . Define π(z) = v and π(f ) = f for all f ∈ C(X) ⊂ C(X × T). This is well-defined because v and f commute in C ∗ (X, α) Z. Moreover, π is an isomorphism from C(X × T) onto its image. We define π(u) = uα . It is easy to check that π(u)π(f )π(u∗ ) = f α −1 = π(f α −1 )
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for all f ∈ C(X) and that π(u)π(z)π(u∗ ) = e−2π
√ −1ξ(α −1 (x))
v = π(e−2π
√ −1ξ(α −1 (x))
z) = π(z ◦ α −1 ).
Therefore π is a homomorphism from A to C ∗ (X, α) Z. Clearly π is surjective. It is also straightforward to see π(g ◦ (id ×Rt )−1 ) = ιˆ(ξ )t (π(g)) for all g ∈ C(X × T) and t ∈ T. It remains to show that π is an isomorphism. Let E be the conditional expectation from A to C(X × T). It is well-known that E is faithful. We can define an action of T on C ∗ (X, α) × Z by √ −1t
γt (f ) = f, γt (uα ) = e2π for t ∈ T. Let
uα , and γt (v) = v
E0 (a) =
T
γt (a) dt
for a ∈ C ∗ (X, α) Z. Then we have π ◦ E = E0 ◦ π . The faithfulness of E leads us to the conclusion.
Remark 4.6. By [Pu1, Corollary 5.7] or [HPS, Theorem 5.5] there exist bijective correspondences between the following spaces. (1) The state space S(K 0 (X, α)) of K 0 (X, α). (2) The tracial state space T (C ∗ (X, α)) of C ∗ (X, α). (3) The set Mα of all α-invariant probability measures. By Proposition 4.5 and Lemma 4.4, (2) and (3) of the above are also identified with the following. (4) The set of ιˆ(ξ )-invariant traces on A = C ∗ (X × T, α × Rξ ). (5) The set of probability measures on X × T which are invariant under α × Rξ and the rotation id ×Rt . Suppose that s ∈ S(K0 (A)) is a state on the ordered group K0 (A). Then there is ν ∈ Mα such that s([f ]) = Sν ([f ]) for all [f ] ∈ K 0 (X, α), where Sν is a state on K 0 (X, α) coming from ν and K 0 (X, α) is viewed as a subgroup of K0 (A). The α-invariant measure ν extends to an α × Rξ -invariant measure on X × T, and so we can extend Sν on K0 (A) (different choices of α × Rξ -invariant measures do not concern the extension of Sν ). For x ∈ K0 (A), S(K 0 (X, α)) Sµ → Sµ (x) ∈ R is an affine function on the state space S(K 0 (X, α)). Since the image of K 0 (X, α) is dense in Aff(S(K 0 (X, α))), for any ε > 0, there exists f1 , f2 ∈ C(X, Z) such that Sµ (x) − ε < Sµ ([f1 ]) < Sµ (x) < Sµ ([f2 ]) < Sµ (x) + ε for all µ ∈ Mα . If A is simple, then it follows from Theorem 3.13 that [f1 ] < x < [f2 ] in K0 (A). In particular, we have Sν (x) − ε < Sν ([f1 ]) = s([f1 ]) < s(x) < s([f2 ]) = Sν ([f2 ]) < Sν (x) + ε. Hence s is equal to Sν as a state on K0 (A). Consequently, the state space S(K 0 (X, α)) can be identified with
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(6) The state space S(K0 (A)) of A when A is simple. Theorem 4.7. Let (X, α) be a Cantor minimal system and let ξ ∈ C(X, T). Suppose that α × Rξ is a minimal homeomorphism on X × T. For the unital simple C ∗ -algebra A = C ∗ (X × T, α × Rξ ), the following conditions are equivalent. (1) α × Rξ is rigid. (2) A has real rank zero. (3) For every extremal tracial state τ on C ∗ (X, α) and every n ∈ N, ι(ξ )n is not weakly inner in the GNS representation πτ . If C ∗ (X, α) has finitely many extremal traces, then the conditions above are also equivalent to (4) ι(ξ ) has the tracial cyclic Rohlin property. Proof. (1)⇔(2) was shown in Corollary 3.11. (1)⇒(3) follows [K, Proposition 2.3]. (3)⇒(1). Suppose that α × Rξ is not rigid. By Lemma 4.4, there exist an ergodic measure µ ∈ Mα , n ∈ N and a Borel function η : X → T such that nξ(x) = η(x) − ηα(x) √
for µ-almost every x ∈ X. Define h ∈ L∞ (X, µ) by h(x) = e2π −1η(x) and let V be the multiplication operator by h on L2 (X, µ). Let τ be the extremal trace on C ∗ (X, α) corresponding to µ. We can regard πτ (C ∗ (X, α)) as a C ∗ -subalgebra of B(L2 (X, µ)). Then, it is not hard to see that V commutes with πτ (f ) for all f ∈ C(X) and that V ∗ πτ (uα )V = πτ (ι(ξ )n (uα )). Namely ι(ξ )n is weakly inner in the GNS representation πτ . (3)⇒(4). From [OP] we can see that ι(ξ ) has the tracial Rohlin property. The conclusion follows from [LO, Theorem 3.4]. (4)⇒(2). It follows from [LO, Theorem 2.9] that A has tracial rank zero. The conclusion is immediate from [L1, Theorem 3.4].
In the theorem above, it was shown that if α × Rξ is rigid and α has only finitely many ergodic measures, then A has tracial rank zero. In the next section we will prove that the hypothesis of finitely many ergodic measures is actually not necessary. 5. Tracial Rank Throughout this section, let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map. We denote the crossed product C ∗ -algebra C ∗ (X × T, α × Rξ ) by A and its implementing unitary by u. We would like to show that if α × Rξ is rigid then A has tracial rank zero. The proof will be done by some improvement of Lemma 3.7. Following the notation used there, we define Ax = C ∗ (C(X × T), uC0 ((X \ {x}) × T)) for x ∈ X. √ Define z ∈ C(X × T) by z(x, t) = e2π −1t . The key step of the proof is approximately unitary equivalence of z1U and z1V in Ax , where U and V are suitable clopen subsets of X satisfying [1U ] = [1V ]. When one uses the fact that Ax has tracial rank zero,
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the proof is just an application of [L6, Theorem 3.4], in which actually a more general result has been obtained. But we would like to include an elementary proof which does not use tracial rank zero for the reader’s convenience. The following lemma says that rigidity implies that the values of a cocycle are uniformly distributed in T. Lemma 5.1. Suppose that α × Rξ is rigid. For any irrational s ∈ T and any ε > 0, there exists N ∈ N such that the following is satisfied: for any n ≥ N and y ∈ X there is a permutation σ on {1, 2, . . . , n} such that |ks −
σ (k)−1
ξ(α i (y))| < ε
i=0
holds for all k ∈ {1, 2, . . . , n}. Proof. For n ∈ N, put ξn (x) =
n−1
ξ(α i (x)). By Lemma 4.4, we have f (t) dν = f (t) dt
i=0
X×T
T
for every invariant measure ν of α × Rξ and every f ∈ C(T). Hence for any f ∈ C(T) and ε > 0 there exists N ∈ N such that
n−1
1
f (ξn (x)) − f (t) dt < ε
n
T i=0
for all n ≥ N and x ∈ X. By a slight modification of [KK, Lemma 2], we can get the conclusion. We leave the details to the reader.
In the following lemmas we need the idea of induced transformations. Let U be a clopen subset of X. Define r : U → N by r(x) = min{n ∈ N : α n (x) ∈ U }. Since α is minimal, r is well-defined and continuous. Put α(y) ˜ = α r(y) (y) for every y ∈ U . Thus α˜ is the first return map on U . It is well-known that (U, α) ˜ is a Cantor minimal system and the associated crossed product C ∗ (U, α) ˜ is canonically identified with 1U C ∗ (X, α)1U . Define ξ˜ : U → T by ξ˜ (y) =
r(y)−1
ξ(α i (y))
i=0
for all y ∈ U . Then α˜ × Rξ˜ is the first return map of α × Rξ on U × T and the associated crossed product C ∗ (U ×T, α˜ ×Rξ˜ ) is identified with 1U ×T A1U ×T . Note that the unitary implementing α˜ × Rξ˜ is given by
un 1Un ×T ,
n∈N
where Un = r −1 (n) and the summation is actually finite.
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In general there is a bijective correspondence between invariant measures of the induced transformation and those of the original one. It follows that α × Rξ is rigid if and only if α˜ × Rξ˜ is rigid. For x ∈ X, let k be the minimal natural number such that α −k (x) ∈ U and set x˜ = α −k (x). Then it is not hard to see that 1U Ax 1U = C ∗ (C(U × T), uC ˜ 0 ((U \ {x}) ˜ × T)). Lemma 5.2. Let α × Rξ be a rigid homeomorphism and let x ∈ X. Suppose that U is a nonempty clopen subset of X. For any s ∈ T and any ε > 0, there exists a unitary w ∈ 1U (Ax ∩ C ∗ (X, α))1U such that wzw∗ − e2π
√ −1s
z1U < ε.
Proof. At first we consider the case U = X. Clearly we may assume that s is irrational. By applying Lemma 5.1 we can find N ∈ N. Let P = {X(v, k) : v ∈ V , k = 1, 2, . . . , h(v)} be a Kakutani-Rohlin partition such that the roof set R(P) contains x and h(v) is greater than N for every v ∈ V . By dividing each tower if necessary, we may assume that P is sufficiently finer so that whenever y1 , y2 ∈ X(v, k) we have |ξ(y1 ) − ξ(y2 )| < ε/ h(v). For every v ∈ V , choose yv ∈ X(v, 1) arbitrarily. By Lemma 5.1 there is a permutation σv on {1, 2, . . . , h(v)} such that |ks −
σv (k)−1
ξ(α i (yv ))| < ε
i=0
for all k ∈ {1, 2, . . . , h(v)}. Put w=
h(v)
1X(v,σv (i)) uσv (i)−σv (i+1) ,
v∈V i=1
where u is the implementing unitary of C ∗ (X, α). It is easily verified that w is a unitary of Ax ∩ C ∗ (X, α). Moreover we get the estimate wzw ∗ − e2π
√ −1s
z < 4π ε.
Let us consider the general case. We follow the notation used in the discussion before the lemma. Applying the first part of the proof to α˜ × ξ˜ and x, ˜ we obtain a unitary w in C ∗ (C(U ), uC ˜ 0 (U \ {x})) ˜ = 1U (Ax ∩ C ∗ (X, α))1U which satisfies the required inequality.
Lemma 5.3. Let α ×Rξ be a rigid homeomorphism and let x ∈ X. For any η ∈ C(X, T) and any ε > 0, there exists a unitary w ∈ Ax ∩ C ∗ (X, α) such that wzw∗ − e2π where z ∈ C(X × T) is given by z(x, t) =
√ −1η
z < ε,
√ e2π −1t .
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Proof. Let P be a partition of X such that whenever y1 , y2 ∈ U ∈ P we have |η(y1 ) − η(y2 )| < ε. For every U ∈ P, by Lemma 5.2, we obtain a unitary wU ∈ 1U (Ax ∩ C ∗ (X, α))1U satisfying ∗ − e2π wU zwU
√ −1η(x) ˜
z1U < ε.
Let w be the product of all wU ’s. Then w is the desired unitary.
Lemma 5.4. Let α × Rξ be a rigid homeomorphism and let x ∈ X. Suppose that U is a clopen neighborhood of x and U, α(U ), . . . , α M (U ) are mutually disjoint. Put p = 1U and q = 1α M (U ) . Then for any ε > 0 there exists a partial isometry w ∈ Ax ∩ C ∗ (X, α) such that w∗ w = p, ww ∗ = q and wzw ∗ − zq < ε. Moreover we have u∗i wui ∈ Ax ∩ C ∗ (X, α) for all i = 0, 1, . . . , M − 1. Proof. There exists a partial isometry v1 ∈ Ax ∩ C ∗ (X, α) such that v1∗ v1 = p and √ v1 v1∗ = q. We have v1∗ zv1 = e2π −1η zp for some continuous function η defined on U . We consider the induced transformation on U . Let α, ˜ ξ˜ and x˜ be as in the discussion before Lemma 5.2. Then Lemma 5.3 applies to them and yields a unitary v2 ∈ p(Ax ∩ C ∗ (X, α))p satisfying v2 zpv2∗ − e2π
√ −1η
zp < ε.
Then w = v1 v2 satisfies wzw ∗ − zq < ε. Since U, α(U ), . . . , α M (U ) are mutually disjoint, one can check that w belongs to Aα i (x) ∩ C ∗ (X, α) for all i = 0, 1, . . . , M − 1. It follows that u∗i wui ∈ Ax ∩ C ∗ (X, α) for all i = 0, 1, . . . , M − 1.
Lemma 5.5. Suppose that α × Rξ is rigid. Let x ∈ X. For any N ∈ N, ε > 0 and a finite subset F ⊂ C(X × T), we can find a natural number M > N, a clopen neighborhood U of x and a partial isometry w ∈ Ax which satisfy the following. (1) α −N +1 (U ), α −N+2 (U ), . . . , U, α(U ), . . . , α M (U ) are mutually disjoint, and µ(U ) < ε/M for all α-invariant measure µ. (2) w ∗ w = 1U and ww ∗ = 1α M (U ) . (3) u∗i wui ∈ Ax for all i = 0, 1, . . . , M − 1. (4) wf − f w < ε for all f ∈ F. Proof. Without loss of generality, we may assume F = {f1 , f2 , . . . , fk , z}, where fi belongs to C(X) ⊂ C(X × T). There exists a clopen neighborhood O of x such that |fi (x) − fi (y)| < ε/2 for all y ∈ O and i = 1, 2, . . . , k. Since α is minimal, we can find M > N such that α M (x) ∈ O. Let U be a clopen neighborhood of x such that the condition (1) is satisfied and U ∪ α M (U ) ⊂ O. Now Lemma 5.4 applies and yields a partial isometry w. It is clear that w is the desired one.
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Theorem 5.6. Suppose that α × Rξ is minimal. Then the following are equivalent. (1) α × Rξ is rigid. (2) A = C(X × T, α × Rξ ) has real rank zero. (3) A = C(X × T, α × Rξ ) has tracial rank zero. (4) A = C(X × T, α × Rξ ) is a unital simple AT-algebra with real rank zero. Proof. It has been proved in Corollary 3.11 that (1) and (2) are equivalent. (3)⇔(4) follows the classification theorem of [L5], since A has torsion free K-theory (see Lemma 2.4). (4)⇒(2) is obvious. (1)⇒(3). We will show the following: For any ε > 0, any finite subset F ⊂ C(X×T) and any nonzero positive element c ∈ A, there exists a projection e ∈ Ax such that the following conditions hold. • ae − ea < ε for all a ∈ F ∪ {u}. • For any a ∈ F ∪ {u}, there exists b ∈ eAx e such that eae − b < ε. • 1 − e is equivalent to a projection in cAc. It follows from Proposition 3.3 that eAx e is a unital simple AT-algebra with real rank zero. Therefore it has tracial rank zero (for example see [L3, Theorem 4.3.5]). Thus, if the above is proved, by [HLX, Theorem 4.8], A has tracial rank zero. In Sect. 3 it was proved that A has real rank zero, stable rank one and has weakly unperforated K0 (A), and so it suffices to show the following: For any ε > 0 and a finite subset F ⊂ C(X ×T), there exists a projection e ∈ Ax such that the following conditions hold. • ae − ea < ε for all a ∈ F ∪ {u}. • For any a ∈ F ∪ {u}, there exists b ∈ eAx e such that eae − b < ε. • τ (1 − e) < ε for all τ ∈ T (A). We may assume F ∗ = F. Choose N ∈ N so that 2π/N is less than ε. Applying Lemma 5.5 to N , ε/2 and a finite subset G=
N−1
ui Fui∗ ,
i=0
we obtain M > N, a clopen neighborhood U of x and a partial isometry w ∈ Ax . Put p = 1U and q = 1α M (U ) . For t ∈ [0, π] we define P (t) = p cos2 t + w sin t cos t + w∗ sin t cos t + q sin2 t. Then P (t) is a continuous path of projections with P (0) = p and P (π ) = q. By the choice of w we obtain the estimate ui∗ P (t)ui f − f ui∗ P (t)ui < ε for all t ∈ [0, π], i = 0, 1, . . . , N − 1 and f ∈ F. We define a projection e by e =1−
M−N i=0
u pu + i
i∗
N−1 i=1
i∗
u P (iπ/N )u
i
.
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The partial isometry w satisfies ui∗ wui ∈ Ax for all i = 1, 2, . . . , N − 1, and so e is a projection of Ax . Evidently we have f e − ef < ε for all f ∈ F. Since P (iπ/N ) − P ((i − 1)π/N ) <
2π < ε, N
it is not hard to see ue − eu < ε. It is clear that ef e belongs to Ax for all f ∈ C(X ×T). It follows from eue = eu(1−p)e that eue also belongs to Ax . We can easily verify τ (1 − e) < Mτ (p) < for all τ ∈ T (A).
ε 2
6. The Generalized Rieffel Projection By Lemma 2.4, the Ki -group (i = 1, 2) of the crossed product C ∗ -algebra arising from an orientation preserving homeomorphism α × ϕ is isomorphic to the direct sum of Z and K 0 (X, α). Needless to say, the equivalence class of the implementing unitary is the generator of Z in the K1 -group. This section is devoted to specify a projection of C ∗ (X × T, α × ϕ) which gives a representative of the generator of the Z-summand of the K0 -group. At first, we consider the case that a cocycle takes its values in the rotation group of the circle. Let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map. We denote C ∗ (X × T, α × Rξ ) by A for short and denote the implementing unitary by u. We will identify K 0 (X, α) as a subgroup of K0 (A). Let ξ˜ ∈ C(X, R) be an arbitrary lift of ξ ∈ C(X, T). Then Mα µ → µ(ξ˜ ) ∈ R gives an affine function on the set of invariant probability measures Mα . The other lifts of ξ are of the form ξ˜ + f with f ∈ C(X, Z), and so this affine function is uniquely determined up to the natural image of K 0 (X, α) in Aff(Mα ). Suppose [ξ ] = 0 in KT0 (X, α). Then there exists η ∈ C(X, T) such that ξ = η − ηα −1 . When η˜ ∈ C(X, R) is a lift of η, ξ˜ = η˜ − ηα ˜ −1 is a lift of ξ and µ(ξ˜ ) = 0 for all µ ∈ Mα . Therefore we obtain a homomorphism : KT0 (X, α) [ξ ] → ([ξ ]) ∈ Aff(Mα )/D(K 0 (X, α)). This homomorphism is known to be surjective ([M1, Lemma 6.2]). Lemma 6.1. Let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map. For any ε > 0, there exists η ∈ C(X, T) such that |(ξ + η − ηα)(x)| < ε for all x ∈ X.
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Proof. Let P = {X(v, k) : v ∈ V , k = 1, 2, . . . , h(v)} −1 be a Kakutani-Rohlin partition of (X, α) such that h(v) > ε for every v ∈ V . We denote the roof set v∈V X(v, h(v)) by R(P). Put
κ(x) =
h(v)
ξ(α k−1 (x))
k=1
for all x ∈ X(v, 1) and v ∈ V . Since X is totally disconnected, there exists a real valued continuous function κ˜ on α(R(P)) such that κ(x) ˜ + Z = κ(x) and − 1 < κ(x) ˜ <1 for all x ∈ α(R(P)) = v∈V X(v, 1). Define η ∈ C(X, T) by η(x) = 0 for all x ∈ α(R(P)) and η(α k (x)) =
k
ξ(α i−1 (x)) −
i=1
k κ(x) ˜ +Z h(v)
for x ∈ X(v, 1), v ∈ V and k = 1, 2, . . . , h(v) − 1. It is not hard to see that η is the desired function.
In a similar fashion to the lemma above, we can show the following, which will be used later. Lemma 6.2. Let (X, α) be a Cantor minimal system and let ξ : X → T and c : X → Z2 be continuous maps. For any ε > 0, there exists η ∈ C(X, T) such that |ξ(x) + η(x) − (−1)c(x) ηα(x)| < ε for all x ∈ X. Definition 6.3. Let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map. Define H (α, ξ ) = {η ∈ C(X, T) : (ξ + η − ηα)(x) ∈ (1/10, 9/10) for all x ∈ X}. By Lemma 6.1, H (α, ξ ) is not empty. Suppose that η belongs to H (α, ξ ). We define a projection e(α, ξ, η) in A as follows. Define a real valued continuous function gη on X × T by √ 10(t − η(x))(1 − 10(t − η(x))) t ∈ [η(x), η(x) + 1/10] gη (x, t) = 0 otherwise. Put η = (ξ + η) ◦ α −1 and define a real valued continuous function f (α, ξ, η) on X × T by 10(t − η(x)) t ∈ [η(x), η(x) + 1/10] 1 t ∈ [η(x) + 1/10, η (x)] f (α, ξ, η)(x, t) = 1 − 10(t − η (x)) t ∈ [η (x), η (x) + 1/10] 0 otherwise.
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Then it is easy to check that e(α, ξ, η) = gη u∗ + f (α, ξ, η) + ugη ∈ A is a self-adjoint projection. We call e(α, ξ, η) the generalized Rieffel projection. Let ξ˜ ∈ C(X, R) be a lift of ξ + η − ηα such that 1/10 < ξ˜ < 9/10. Then, for every µ ∈ Mα , τµ (e(α, ξ, η)) = τµ (f (α, ξ, η)) = µ(ξ˜ ), where τµ is the tracial state on A corresponding to µ. Hence the affine function µ → τµ (e(α, ξ, η)) is a representative of ([ξ ]) ∈ Aff(Mα )/D(K 0 (X, α)). Proposition 6.4. In the situation above, let e = e(α, ξ, η) ∈ A be the generalized Rieffel projection. Then K 0 (X, α) and [e] generate K0 (A). Proof. Let v the unilateral shift on 2 (N) and let T be the Toeplitz algebra generated by v. Put q = 1 − vv ∗ . In the C ∗ -algebra A ⊗ T , we consider the C ∗ -subalgebra B generated by C(X × T) ⊗ 1 and u ⊗ v ∗ . There is a surjective homomorphism π from B to A sending u ⊗ v ∗ to u. The kernel of π is C(X × T) ⊗ K, where K is the algebra of compact operators. Put a = gη √ u∗ ⊗ v + f (α, ξ, η) ⊗ 1 + ugη ⊗ v ∗ . Then π(a) = e. 2π −1a is a generator of K (C1 ⊗ C(T)) ∼ Z. Define Hence it suffices to show that e = 1 X a continuous function h(t) by √ h(t) = (1 − 8t (1 − t)) + −1t (t − 1)(t − 1/2). √
Since e2π −1t is homotopic to h(t) in the set of complex-valued invertible functions, it suffices to know the K1 -class of the invertible element h(a). By a − a 2 = gη2 ⊗ q and (a 2 − a)(a − 1/2) = −(gη2 (f (α, ξ, η) − 1/2)) ⊗ q, it follows that h(a) is homotopic to 1X ⊗ z ⊗ q in GL(C(X × T)) ⊗ q, where z(t) = √ e2π −1t is the generator of K1 (C(T)).
Let K0 (A) ∼ = Z ⊕ K 0 (X, α) be the isomorphism described in the proposition above. If α × Rξ is minimal, then it follows from Theorem 3.13 that K0 (A)+ ∼ = {(n, [f ]) : µ(nξ˜ + f ) > 0 for all µ ∈ Mα } ∪ {0}. See also Remark 4.6. We now turn to the general case. Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo+ (T) be a continuous map. We write the crossed product C ∗ -algebra arising from (X × T, α × ϕ) by A and the implementing unitary by u. In order to define the Rieffel projection in A, we need some preparations. For ϕ ∈ Homeo+ (T), let r(ϕ) ∈ T denote the rotation number. The reader may refer to [KH, Chapter 11] for the definition and some elementary properties of r(ϕ). Lemma 6.5. Let ϕ ∈ Homeo+ (T). The map T t → r(Rt ϕ) is a continuous surjection from T to T of degree one.
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Proof. By the definition of the rotation number, we see that the mapping t → r(Rt ϕ) is nondecreasing as a real valued function. It is clear that r(Rt ϕ) = 0 if and only if t belongs to I = {s − ϕ(s) : s ∈ T}. Since ϕ is an orientation preserving homeomorphism, I is not the whole circle. Thus, I is a closed interval of the circle. It follows that the map is a surjection of degree one.
Lemma 6.6. Let X be the Cantor set and let I ⊂ T be an open subset. When : X × T → T is a continuous map and (x, ·) is surjective for every x ∈ X, there exists a continuous map ξ : X → T such that (x, ξ(x)) ∈ I for all x ∈ X.
Proof. By assumption, for each x ∈ X, there exists tx ∈ T such that (x, tx ) ∈ I . The continuity of implies that there exists a clopen neighborhood Ux of x such that (Ux , tx ) ⊂ I . Since X is compact, it is covered by finitely many Ux ’s. We can find a locally constant function ξ ∈ C(X, T) satisfying the required property.
The following lemma corresponds to Lemma 6.1. Lemma 6.7. Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo+ (T) be a continuous map. Suppose that an open subset I ⊂ T is given. Then there exists η ∈ C(X, T) such that −1 r(Rηα(x) ◦ ϕx ◦ Rη(x) )∈I
for all x ∈ X.
Proof. Define (x, t) = r(Rt ϕx ). It is obvious that is continuous. By Lemma 6.5, for each x ∈ X, (x, ·) is a continuous surjection. It follows from Lemma 6.6 that there exists a continuous map ξ : X → T such that (x, ξ(x)) ∈ I for all x ∈ X. Moreover, there exists ε > 0 such that, for all x ∈ X, if |t − ξ(x)| < ε then (x, t) ∈ I . From Lemma 6.1, we can find η ∈ C(X, T) such that |(ξ + η − ηα)(x)| < ε for all x ∈ X. Then we get −1 ) = r(Rη(x) ◦ Rηα(x)−η(x) ◦ ϕx ◦ R−η(x) ) r(Rηα(x) ◦ ϕx ◦ Rη(x)
= r(Rηα(x)−η(x) ◦ ϕx ) = (x, ηα(x) − η(x)) ∈ I, thereby completing the proof.
By the lemma above, without loss of generality, we may always assume that r(ϕx ) is not zero for all x ∈ X. Put c = inf{|ϕx (t) − t|, |ϕx−1 (t) − t| : (x, t) ∈ X × T}.
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Since ϕx has no fixed points, c is a positive real number. Take s ∈ T. We define a function f (α, ϕ, s) ∈ C(X × T) by −1 c (t − s) t ∈ [s, s + c] 1 t ∈ [s + c, ϕα −1 (x) (s)] f (α, ϕ, s)(x, t) = 1 − c−1 (ϕα−1 t ∈ [ϕα −1 (x) (s), ϕα −1 (x) (s + c)] −1 (x) (t) − s) 0 otherwise. By the choice of c, f is well-defined. Define a function gs ∈ C(X × T) by c−1 (t − s)(1 − c−1 (t − s)) t ∈ [s, s + c] gs (x, t) = 0 otherwise. Then one checks that e(α, ϕ, s) = gs u∗ + f (α, ϕ, s) + ugs is a well-defined projection of A. Let us call it the generalized Rieffel projection for A. In exactly the same way as Proposition 6.4, we can show the following. Proposition 6.8. In the above setting, K0 (A) is generated by K 0 (X, α) and [e(α, ϕ, s)]. Furthermore, T s → e(α, ϕ, s) ∈ A is a continuous path of projections in A. In the definition of e(α, ϕ, s), we can replace gs by zgs , where z is a complex number with |z| = 1. But, this choice does not matter to the homotopy equivalence class of the projection. 7. Approximate K-Conjugacy Let us begin with recalling the definition of weakly approximate conjugacy. Definition 7.1 ([LM, Definition 3.1]). Let (X, α) and (Y, β) be dynamical systems on compact metrizable spaces X and Y . We say that (X, α) and (Y, β) are weakly approximately conjugate, if there exist homeomorphisms σn : X → Y and τn : Y → X such that σn ασn−1 converges to β in Homeo(Y ) and τn βτn−1 converges to α in Homeo(X). Here we use the topology on Homeo(X) defined in Sect. 2. In other words, in the above definition, we require that lim g ◦ σn α −1 σn−1 − g ◦ β −1 = 0 and
n→∞
lim f ◦ τn β −1 τn−1 − f ◦ α −1 = 0
n→∞
for all g ∈ C(Y ) and f ∈ C(X). In [LM, Theorem 4.13], it was shown that two Cantor minimal systems are weakly approximately conjugate if and only if they have the same periodic spectrum (see also [M3, Theorem 3.1]). Similar results were shown in [M3] for dynamical systems on the product of the Cantor set and the circle. As one sees that in the above definition, there is no consistency among σn or τn . It is clear (see [LM]) that the relation can be made stronger if one requires some consistency among σn as well as τn . We hope such a stronger version of approximate conjugacy is a more reasonable replacement of (flip) conjugacy. Suppose σn ασn−1 → β in Homeo(Y ). In [LM, Proposition 3.2], it was shown that there exists an asymptotic morphism {ψn } : B → A such that lim ψn (f ) − f ◦ σn = 0
n→∞
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for all f ∈ C(Y ) and lim ψn (uβ ) = uα ,
n→∞
where uα and uβ denote the implementing unitaries in C ∗ (X, α) and C ∗ (Y, β). This observation, however, is far from the existence theorem in classification theory, which requires that {ψn } carries an isomorphism of K-groups (see [L5, Theorem 4.3] for instance). As pointed out in [LM], we have to impose conditions on the conjugating maps σn so that the associated asymptotic morphism has a nice property. Taking account of this, we make the following definitions. By an order and unit preserving homomorphism ρ : K∗ (B) → K∗ (A), we mean a pair of homomorphisms ρi : Ki (A) → Ki (B) (i = 0, 1) such that ρ0 ([1A ]) = [1B ] and ρ0 (K0 (A)+ ) ⊂ K0 (B)+ . Definition 7.2. Let (X, α) and (Y, β) be dynamical systems on compact metrizable spaces X and Y . Suppose that a sequence of homeomorphisms σn : X → Y satisfies σn ασn−1 → β in Homeo(Y ). Let {ψn } be the asymptotic morphism arising from σn . We say that the sequence {σn } induces an order and unit preserving homomorphism ρ : K∗ (C ∗ (Y, β)) → K∗ (C ∗ (X, α)) between K-groups, if for every projection p ∈ M∞ (C ∗ (Y, β)) and every unitary u ∈ M∞ (C ∗ (Y, β)), there exists N ∈ N such that [ψn (p)] = ρ([p]) ∈ K0 (C ∗ (X, α)) and [ψn (u)] = ρ([u]) ∈ K1 (C ∗ (X, α)) for every n ≥ N. Definition 7.3 ([L6, Definition 5.3]). Let (X, α) and (Y, β) be dynamical systems on compact metrizable spaces X and Y . We say that (X, α) and (Y, β) are approximately K-conjugate, if there exist homeomorphisms σn : X → Y , τn : Y → X and an isomorphism ρ : K∗ (C ∗ (Y, β)) → K∗ (C ∗ (X, α)) between K-groups such that σn ασn−1 → β, τn βτn−1 → α and the associated asymptotic morphisms {ψn } : B → A and {ϕn } : A → B induce the isomorphisms ρ and ρ −1 . We say that (X, α) and (Y, β) are approximately flip K-conjugate, if (X, α) is approximately K-conjugate to either of (Y, β) and (Y, β −1 ). J. Tomiyama [T] proved that two topological transitive systems (X, α) and (Y, β) are flip conjugate if and only if there is an isomorphism ϕ : C ∗ (X, α) → C ∗ (Y, β) such that ϕ ◦ jα = jβ ◦ χ for some isomorphism χ : C(X) → C(Y ). Definition 7.4 ([L6, Definition 3.8]). Let (X, α) and (Y, β) be two topological transitive systems. We say that (X, α) and (Y, β) are C ∗ -strongly approximately flip conjugate if there exists a sequence of isomorphisms ϕn : C ∗ (X, α) → C ( Y, β) and a sequence of isomorphisms χn : C(X) → C(Y ) such that [ϕn ] = [ϕ1 ] in KL(C ∗ (X, α), C ∗ (Y, β)) for all n ∈ N and lim ϕn ◦ jα (f ) − jβ ◦ χn (f ) = 0
n→∞
for all f ∈ C(X).
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Let X be the Cantor set and let ξ be a continuous function from X to T. In this section, we would like to discuss approximate K-conjugacy for (X × T, α × Rξ ). As in the previous section, let A denote the crossed product C ∗ -algebra C ∗ (X × T, α × Rξ ). By Lemma 6.1, we can find a continuous function ζ : X → T such that [ζ ] = [ξ ] in KT0 (X, α) and ζ (x) ∈ (7/15, 8/15) for every x ∈ X. By Lemma 4.3, α ×Rξ is conjugate to α × Rζ . Therefore we may assume ξ(x) ∈ (7/15, 8/15) for all x ∈ X without loss of generality. Suppose that η belongs to H (α, ξ ). Let η˜ ∈ C(X, R) be a lift of η. Since (ξ + η − ηα)(x) ∈ (1/10, 9/10) for every x ∈ X, we have (η − ηα)(x) ∈ / [13/30, 17/30]. Hence there exists a unique f ∈ C(X, Z) such that f − (η˜ − ηα) ˜ ∞ <
1 . 2
It is easy to see that [f ] ∈ K 0 (X, α) is independent of the choice of η. ˜ Let us denote [f ] by Bα (η). Lemma 7.5. In the above setting, suppose that η and η are homotopic in H (α, ξ ). (1) e(α, ξ, η) is homotopic to e(α, ξ, η ) in the set of projections of A. (2) Bα (η) = Bα (η ). Proof. (1) Suppose [0, 1] t → κt ∈ H (α, ξ ) is a homotopy from η to η . Then the generalized Rieffel projection e(α, ξ, κt ) is well-defined and t → e(α, ξ, κt ) gives a continuous path of projections in A from e(α, ξ, η) to e(α, ξ, η ). (2) Let κt be a homotopy as in (1). There exists a continuous map κ˜ from X × [0, 1] to R such that κ˜ t (x) + Z = κt (x) for all x ∈ X and t ∈ [0, 1]. Since (κ˜ t − κ˜ t α)(x) =
1 , 2
the integer nearest to (κ˜ t − κ˜ t α)(x) does not vary as t varies. Hence we get Bα (η) = Bα (η ).
√
Let η ∈ H (α, ξ ). The unitary vη (x) = e2π −1η(x) of C(X) satisfies uα vη u∗α vη∗ − 1∞ < 2, where uα denotes the implementing unitary of C ∗ (X, α), because of (η − ηα)(x) = 1/2. Thus, uα and vη are almost commuting unitaries in a sense. When u, v ∈ Mn (C) are unitaries satisfying uv − vu < 2, on account of det(uvu∗ v ∗ ) = 1, we have 1 Tr(log(uvu∗ v ∗ )) ∈ Z ∼ √ = K0 (Mn (C)), 2π −1 where Tr is the standard trace on Mn (C) and log is the logarithm with values in {z : (z) ∈ (−π, π )}. The Bott element for pairs of almost commuting unitaries in a unital C ∗ -algebra is a generalization of this (see [EL]). More precisely, if u and v are unitaries in a unital C ∗ -algebra and uv − vu ≈ 0, then a projection B(u, v) and an element of the K0 -group are obtained. Our Bα (η) is just this K0 -class for uα and vη .
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Lemma 7.6. Suppose (ξ + η − ηα)(x) ∈ (1/3, 2/3) for all x ∈ X. Then we have [e(α, ξ, η)] = [e(α, ξ, 0)] − Bα (η) in K0 (A). Proof. Put 1 c = inf{|(ξ + η − ηα)(x)| : x ∈ X} − . 3 Then c is positive. Choose a sufficiently finer Kakutani-Rohlin partition P = {X(v, k) : v ∈ V , 1 ≤ k ≤ h(v)} for (X, α). Let R(P) be the roof set of P. We may assume that h(v) is large and sup{|η(x) − η(y)| : x, y ∈ U } <
c 2
where for every U ∈ P, = {X(v, k) : v ∈ V , 1 ≤ k < h(v)} ∪ {R(P)}. P and define η ∈ C(X, Z) by η (x) = η(xU ) for x ∈ U . It is Take xU ∈ U for all U ∈ P not hard to see that η and η are homotopic in H (α, ξ ). By Lemma 7.5, [e(α, ξ, η)] = [e(α, ξ, η )] and Bα (η) = Bα (η ). Hence, by replacing η by η , we may assume that η is Furthermore, by adding a constant function, constant on each clopen set belonging to P. we may assume that η(x) = 0 for all x ∈ R(P). Note that |(η − ηα)(x)| is less than 2/3 − 7/15 = 1/5 for all x ∈ X. There is a unique lift η˜ ∈ C(X, R) of η such that η(x) ˜ = 0 for all x ∈ R(P) and |(η˜ − ηα)(y)| ˜ < 1/5 for all y ∈ R(P)c . Moreover, for every v ∈ V , there exists an integer mv such that 1 5
˜ < |mv − η(x)| for all x ∈ X(v, 1). Therefore Bα (η) is equal to |mv | = |mv − η(α ˜
h(v)−1
(x))| ≤ |mv − η(x)|+ ˜
v∈V
h(v)−1
−mv [1X(v,1) ], and
|η(α ˜ k−1 (x)) − η(α ˜ k (x))| <
k=1
h(v) , 5
where x is a point in X(v, 1). Fix v0 ∈ V . Let a : {1, 2, . . . , h(v0 )} → R be a map such that a(h(v0 )) = 0, |mv0 − a(1)| < 11/30 and |a(k) − a(k + 1)| < 11/30 for all k = 1, 2, . . . , h(v0 ) − 1. Define a continuous map κ from X to R by a(k) x ∈ X(v0 , k), k = 1, 2, . . . , h(v0 ) κ(x) = η(x) ˜ otherwise. Put κ(x) ˆ = κ(x) + Z. Then κˆ ∈ C(X, T) belongs to H (α, ξ ), because 7/15 − 11/30 = 1/10. For t ∈ [0, 1], put κt = tκ + (1 − t)η. ˜ Then it is not hard to see that κˆt gives a homotopy from η to κˆ in H (α, ξ ).
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At first, let us consider the case that mv0 is positive. Define a : {1, 2, . . . , h(v0 )} → R by a(1) = a(2) = mv0 + 13 , a(3) = mv0 and a(k) =
mv0 (h(v0 ) − k) h(v0 ) − 3
for every k = 4, 5, . . . , h(v0 ). By using this map a, define κ ∈ C(X, R) as above. It follows that η and κˆ are homotopic in H (α, ξ ). Let f1 be the continuous function on X × T defined by 10(t − 1/3) (x, t) ∈ X(v0 , 2) × [1/3, 1/3 + 1/10] 1 (x, t) ∈ X(v0 , 2) × [1/3 + 1/10, 2/3] f1 (x, t) = 1 − 10(t − 2/3) (x, t) ∈ X(v0 , 2) × [2/3, 2/3 + 1/10] 0 otherwise. Let U = X(v0 , 2) × T and put f2 = (1U − f1 ) ◦ (α × Rξ )−1 . Define a continuous function g on X × T by √ 10(t − 1/3)(1 − 10(t − 1/3)) (x, t) ∈ X(v0 , 2)× ∈ [1/3, 1/3+1/10] − √ g(x, t) = 10(t − 2/3)(1 − 10(t − 2/3)) (x, t) ∈ X(v0 , 2)× ∈ [2/3, 2/3+1/10] 0 otherwise. Then it can be verified that e = gu∗ + (f1 + f2 ) + ug is a projection and e is equivalent to 1U : in fact, if h ∈ C(X × T) is a function with h|X(v0 , 2) × [1/3, 1/3 + 1/10] = −1, h|X(v0 , 2) × [2/3, 2/3 + 1/10] = 1 and |h|2 = 1, then the partial isometry w = h f1 + 1 U − f 1 u ∗ satisfies w ∗ w = e and ww ∗ = 1U . Furthermore e is a subprojection of e(α, ξ, κ) ˆ and e(α, ξ, κ) ˆ − e = e(α, ξ, η ) for some η ∈ H (α, ξ ) with Bα (η ) = Bα (η) + [1X(v0 ,1) ]. Hence ˆ − Bα (η) [e(α, ξ, 0)] − [e(α, ξ, η)] − Bα (η) = [e(α, ξ, 0)] − [e(α, ξ, κ)] = [e(α, ξ, 0)] − [e(α, ξ, η )] − [e] − Bα (η) = [e(α, ξ, 0)] − [e(α, ξ, η )] − Bα (η ). We can repeat the same argument with η in place of η. By repeating this mv0 times, we will obtain [e(α, ξ, 0)] − [e(α, ξ, η)] − Bα (η) = [e(α, ξ, 0)] − [e(α, ξ, η )] − Bα (η ) with Bα (η ) = v=v0 −mv [1X(v,1) ] and η (x) = 0 for all x ∈ X(v0 , 1) ∪ · · · ∪ X(v0 , h(v0 )). When mv0 is negative, there exists κˆ homotopic to η in H (α, ξ ) such that e(α, ξ, κ) ˆ + e = e(α, ξ, η ) for some η ∈ H (α, ξ ) with Bα (η ) = Bα (η) − [1X(v0 ,1) ]. In a similar fashion to the preceding paragraph, the same conclusion follows. By applying the same argument to all towers in V , we have [e(α, ξ, 0)] − [e(α, ξ, η)] − Bα (η) = 0.
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Lemma 7.7. Let (X, α) be a Cantor minimal system. Suppose that ξ1 , ξ2 ∈ C(X, R) and f ∈ C(X, Z) satisfy µ(ξ2 ) = µ(ξ1 ) + µ(f ) for every α-invariant measure µ ∈ Mα and 7 8 < ξi (x) < 15 15 for all x ∈ X and i = 1, 2. Put ξˆi (x) = ξi (x) + Z for i = 1, 2. Then, for any ε > 0, there exists η ∈ C(X, T) such that |(ξˆ1 − ξˆ2 )(x) − (η − ηα)(x)| < ε for all x ∈ X and Bα (η) = [f ]. Proof. We may assume ε < 10−1 . In the same way as in [GW, Lemma 2.4], we can find a Kakutani-Rohlin partition P = {X(v, k) : v ∈ V , k = 1, 2, . . . , h(v)} such that
h(v)
1
k−1
<ε (ξ − ξ + f )(α (x)) 1 2
h(v)
k=1
for all v ∈ V and x ∈ X(v, 1). Define a real valued continuous function κ on α(R(P)) by κ(x) =
h(v)
(ξ1 − ξ2 + f )(α k−1 (x))
k=1
for x ∈ X(v, 1). Let η be a real valued continuous function on X such that η(α k (x)) =
k
(ξ2 − ξ1 )(α i−1 (x)) +
i=1
k κ(x) h(v)
for all x ∈ X(v, 1) and k = 0, 1, . . . , h(v) − 1. Define η(x) ˆ = η(x) + Z. It is straightforward to check |(ξˆ1 − ξˆ2 )(x) − (ηˆ − ηα)(x)| ˆ <ε for all x ∈ X. For x ∈ / R(P), we have 1 1 + , 15 10 and so the integer nearest to (η − ηα)(x) is zero. If x belongs to the roof set R(P), then |(η − ηα)(x)| <
|(η − ηα)(x) −
h(v)
f (α 1−k (x))| <
k=1
Hence we can conclude that Bα (η) is equal to [f ].
1 1 + . 15 10
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Now we are ready to prove the main theorem of this section. Let (X, α) and (Y, β) be Cantor minimal systems and let ξ : X → T and ζ : Y → T be continuous functions. Suppose that α × Rξ and β × Rζ are both minimal. We denote by A (resp. B) the crossed product C ∗ -algebra arising from (X × T, α × Rξ ) (resp. (Y × T, β × Rζ )). Theorem 7.8. The following are equivalent. (1) (X × T, α × Rξ ) is approximately K-conjugate to (Y × T, β × Rζ ). (2) There exists a unital order isomorphism ρ from K0 (B) to K0 (A) such that ρ(K 0 (Y, β)) = K 0 (X, α). Proof. (1)⇒(2). This is immediate from the definition of approximate K-conjugacy (Definition 7.3). (2)⇒(1). By Lemma 6.1, without loss of generality, we may assume ξ(x), ζ (y) ∈ (7/15, 8/15) for all x ∈ X and y ∈ Y . Let e(α, ξ, 0) ∈ A and e(β, ζ, 0) ∈ B be the projections described in the previous section. We follow the notation used there. Since ρ is an isomorphism, there are only two possibilities: ρ([e(β, ζ, 0)]) − [e(α, ξ, 0)] ∈ K 0 (X, α) or ρ([e(β, ζ, 0)]) + [e(α, ξ, 0)] ∈ K 0 (X, α). Suppose that the latter equality holds. The dynamical system (X × T, α × Rξ ) is conjugate to (X × T, α × R−ξ ) via the mapping (x, t) → (x, −t). This conjugacy induces an isomorphism between the corresponding C ∗ -algebras, which in turn yields an isomorphism between the K0 -groups. One can see that [e(α, ξ, 0)] is sent to [1 − e(α, −ξ, 0)] by this isomorphism. For this reason, by replacing α × Rξ by α × R−ξ , we may always assume that there exists h ∈ C(X, Z) such that ρ([e(β, ζ, 0)]) = [e(α, ξ, 0)] + [h]α in K0 (A). The restriction of ρ to K 0 (Y, β) is a unital order isomorphism onto K 0 (X, α). By [LM, Theorem 5.4] or [M3, Theorem 3.4], there exist homeomorphisms σn : X → Y such that σn ασn−1 → β in Homeo(Y ) and [f σn ]α = ρ([f ]β ) for all f ∈ C(Y, Z) and n ∈ N. We may assume |ζ ◦ β −1 σn (x) − ζ ◦ σn α −1 (x)| <
1 n
for all x ∈ X and n ∈ N. As in Remark 4.6, we denote the state on the K0 -group arising from an invariant measure µ by Sµ . Then, for any µ ∈ Mα and n ∈ N, we have ρ ∗ (Sµ )([f ]β ) = Sµ (ρ([f ]β )) = µ([f σn ]α ) = Sσn∗ (µ) ([f ]β )
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for all f ∈ C(Y, Z). Thus, ρ ∗ (Sµ ) = Sσn∗ (µ) on K 0 (Y, β), and so on K0 (B) (see Remark 4.6). Let ξ˜ ∈ C(X, R) and ζ˜ ∈ C(Y, R) be the lifts of ξ and ζ satisfying 8 7 8 7 < ξ˜ (x) < and < ζ˜ (y) < 15 15 15 15 for all x ∈ X and y ∈ Y . It follows that µ(ζ˜ σn ) = σn∗ (µ)(ζ˜ ) = Sσn∗ (µ) ([e(β, ζ, 0)]) = ρ ∗ (Sµ )([e(β, ζ, 0)]) = Sµ (ρ([e(β, ζ, 0)]) = Sµ ([e(α, ξ, 0)] + [h]α ) = µ(ξ˜ ) + µ(h) for every µ ∈ Mα and n ∈ N. Now Lemma 7.7 applies and yields ηn ∈ C(X, T) such that |(ξ − ζ σn )(x) − (ηn − ηn α)(x)| <
1 n
for all x ∈ X and Bα (ηn ) = [h]. Hence it is easy to verify that (σn × Rηn )(α × Rξ )(σn × Rηn )−1 → (β × Rζ ) in Homeo(Y × T). Furthermore, since 2 n
|ζβ −1 σn (x) − (ξ α −1 + ηn − ηn α −1 )(x)| < for all x ∈ X, we get the estimate f (β, ζ, 0) ◦ (σn × Rηn ) − f (α, ξ, −ηn ) <
20 . n
See the discussion before Proposition 6.4 for the definition of f (·, ·, ·). It is easy to see g0 ◦ (σn × Rηn ) = g−ηn . Therefore, the asymptotic morphism {ψn } : B → A associated with σn × Rηn satisfies lim ψn (e(β, ζ, 0)) − e(α, ξ, −ηn ) = 0,
n→∞
and Lemma 7.6 yields [e(α, ξ, −ηn )] = [e(α, ξ, 0)] + Bα (ηn ) = [e(α, ξ, 0)] + [h]α = ρ([e(β, ζ, 0)]). For every clopen set U ⊂ Y , we know lim ψn (1U ) − 1U ◦ σn = 0
n→∞
and [1U ◦ σn ]α = ρ([1U ]β ). It follows that {ψn } induces ρ : K0 (B) → K0 (A). For every clopen set U ⊂ Y , we know lim ψn (z1U ) − z1U ◦ (σn × Rηn ) = 0,
n→∞
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where z is a unitary defined by z(y, t) = e2π
√ −1t .
It is clear that
[1U c + z1U ◦ (σn × Rηn )] = [1U c + z(1U ◦ σn )] in K1 (A). Since {ψn } approximately carries the implementing unitary of β × Rζ to that of α × Rξ , we can conclude that {ψn } induces an isomorphism between K1 (B) and K1 (A). Consequently {ψn } induces an isomorphism between K-groups. Similarly, we can construct an asymptotic morphism from A to B which induces ρ −1 between their Kgroups.
Theorem 7.9. Suppose that α×Rξ and β ×Rζ are minimal and rigid. Then the following are equivalent. (1) (X × T, α × Rξ ) is approximately K-conjugate to (Y × T, β × Rζ ). (2) There exists a unital order isomorphism ρ from K0 (B) to K0 (A) such that ρ(K 0 (Y, β)) = K 0 (X, α). (3) (X × T, α × Rξ ) is approximately flip K-conjugate to (Y × T, β × Rζ ). (4) α × Rξ and β × Rζ are C ∗ -strongly approximately flip conjugate. (5) There is θ ∈ KL(A, B) which gives an order and unit preserving isomorphism from (K0 (A), K0 (A)+ , [1A ], K1 (A)) onto (K0 (B), K0 (B)+ , [1B ], K1 (B)) and an isomorphism χ : C(X × T) → C(Y × T) such that [jα×Rξ ] × θ = [jβ×Rζ ◦ χ ] in KL(C(X × T), B). Proof. We have seen (1)⇔(2) in Theorem 7.8. (1)⇒(3) is obvious. When both α × Rξ and β × Rζ are rigid, A and B have tracial rank zero by Theorem 5.6. Thus,(3)⇒(4) follows from [L6, Theorem 5.4]. (4)⇒(5) follows immediately from [L6, Theorem 3.9]. (5)⇒(2). Suppose that θ induces an order and unit preserving isomorphism (θ ) from (K0 (A), K0 (A)+ , [1A ], K1 (A)) to (K0 (B), K0 (B)+ , [1B ], K1 (B)). Suppose that there is χ : C(X × T) → C(Y × T) such that [jα×Rξ ] × θ = [jβ×Rζ ◦ χ ]. This implies that (θ ) gives an isomorphism from K 0 (X, α) onto K 0 (Y, β). So (2) holds.
8. Non-Orientation Preserving Case Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuous map. In this section, we would like to consider the case that α × ϕ is not orientation preserving, that is, [o(ϕ)] is not zero in K 0 (X, α)/2K 0 (X, α). As was seen in the discussion before Lemma 2.5, the skew product extension (X × Z2 , α × o(ϕ)) is a Cantor minimal system. This system will play an important role when we study α × ϕ. Define a continuous map ϕ : X × Z2 → Homeo(T)+ by ϕ(x,k) = λk+o(ϕ)(x) ϕx λk , where λ is given by λ(t) = −t for t ∈ T. Let π be the projection from X × Z2 to the first coordinate.
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Lemma 8.1. As a Homeo(T)-valued cocycle on the Cantor minimal system (X×Z2 , α× o(ϕ)), ϕπ is cohomologous to ϕ . In particular, ϕπ is orientation preserving with respect to the minimal homeomorphism α × o(ϕ). Proof. Put ω(x.k) = λk . Then ϕ(x,k) ◦ ω(x,k) = λk+o(ϕ)(x) ◦ ϕx = ω(α×o(ϕ))(x,k) ◦ ϕπ(x,k)
implies that they are cohomologous.
We remark that the following diagram of factor maps is commutative: π×id
(X × T, α × ϕ) ←−−−− (X × Z2 × T, α × o(ϕ) × ϕπ ) (X, α)
←−−−− π
(X × Z2 , α × o(ϕ)).
From Lemma 6.5 and 6.6, there exists ξ ∈ C(X, T) such that r(Rξ(x) λo(ϕ)(x) ϕx ) = 0 for all x ∈ X. By applying Lemma 6.2 to −ξ and o(ϕ), we obtain η ∈ C(X, T) such that |η(x) − (−1)o(ϕ)(x) ηα(x) − ξ(x)| < ε, where ε is sufficiently small so that it implies 0 = r Rη(x)−(−1)o(ϕ)(x) ηα(x) λo(ϕ)(x) ϕx −1 o(ϕ)(x) = r R(−1) ϕx Rη(x) o(ϕ)(x) ηα(x) λ −1 = r λo(ϕ)(x) Rηα(x) ϕx Rη(x) for all x ∈ X. Therefore, by perturbing ϕ by Rη , we may assume r( ϕ(x,k) ) = (−1)k r λo(ϕ)(x) ϕx = 0 for all (x, k) ∈ X × Z2 . Let A denote the crossed product C ∗ -algebra arising from (X × T, α × ϕ). We write the implementing unitary by u. Define an automorphism θ ∈ Aut(A) of order two by θ (f ) = f for all f ∈ C(X × T) and θ (u) = ug, where g ∈ C(X × T) is given by g(x, t) = (−1)o(ϕ)(x) . By Lemma 6.1, it can be seen that θ is approximately inner, and so it induces the identity on the K-group. Proposition 8.2. In the situation above, the crossed product C ∗ -algebra arising from the dynamical system (X × Z2 × T, α × o(ϕ) × ϕπ ) is isomorphic to A θ Z2 .
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Proof. We write the implementing unitary in A by uA . Let us denote C ∗ (X × Z2 × T, α × o(ϕ) × ϕπ ) by B and the implementing unitary in B by uB . Let v denote the unitary which implements θ . We would like to define a homomorphism from A Z2 to B. The C ∗ -algebra A can be naturally embedded into B via the factor map π × id from X × Z2 × T to X × T. Let |A be this embedding. Define a continuous function h ∈ C(X × Z2 × T) by h(x, k, t) = (−1)k . Then, h ◦ (α × o(ϕ) × ϕπ ) = h(g ◦ (π × id)). Put (v) = h. For f ∈ C(X × T), we have h(f )h = h(f ◦ (π × id))h = f ◦ (π × id) = (θ (f )). Besides, h(uA )h = huB h = uB (g ◦ (π × id)) = (θ(uA )). It follows that is a well-defined homomorphism. It is not hard to see that is an isomorphism.
We freely use the identification of the two C ∗ -algebras established in the proposition above. By Lemma 8.1 and 2.4, we know that K0 (A Z2 ) ∼ = Z ⊕ K 0 (X × Z2 , α × o(ϕ)) and K1 (A Z2 ) ∼ = Z ⊕ K 0 (X × Z2 , α × o(ϕ)). On the crossed product C ∗ (X × Z2 × T, α × o(ϕ) × ϕπ ), the dual action θˆ is given by θˆ (f )(x, k, t) = f (x, k + 1, t) for f ∈ C(X × Z2 × T) and θˆ (u) = u, where u is the implementing unitary. Define a homeomorphism γ ∈ Homeo(X × Z2 ) by γ (x, k) = (x, k + 1). Then γ × id commutes with α × o(ϕ) × ϕπ and θˆ on C(X × Z2 × T) is induced by γ × id. Let us consider the induced action θˆ∗ on the K-groups. Evidently, θˆ∗0 on K 0 (X × Z2 , α × o(ϕ)) is given by [f ] → [f γ ], and θˆ∗1 on K 0 (X × Z2 , α × o(ϕ)) is given by [f ] → [−f γ ].
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Of course, θˆ∗ ([u]) = [u]. Hence it remains to know the image of the generalized Rieffel projection. Let e = e(α × o(ϕ), ϕ , 0) = g0 u∗ + f (α × o(ϕ), ϕ , 0) + ug0 be the projection of C ∗ (X × Z2 × T, α × o(ϕ) × ϕ ) as in Proposition 6.8. This is well-defined, because r( ϕ(x,k) ) is not zero. By the map (x, k, t) → (x, k + 1, λ(t)), f (α × o(ϕ), ϕ , 0) is carried to a function supported on (x, k, t) : t ∈ [− ϕ(α×o(ϕ))−1 (x,k+1) (c), 0] and g0 is carried to a function supported on X × Z2 × [−c, 0]. Note that − ϕ(α×o(ϕ))−1 (x,k+1) (c) = −λk+1 ϕα −1 (x) λk+1+o(ϕ)(α = −λ ϕ(α×o(ϕ))−1 (x,k) λ(c) = ϕ(α×o(ϕ))−1 (x,k) (−c).
−1 (x))
(c)
Hence, under the identification of C ∗ (X × Z2 × T, α × o(ϕ) × ϕ ) with C ∗ (X × Z2 × T, α × o(ϕ) × ϕπ ), we have θˆ (e) = θˆ (e(α × o(ϕ), ϕ , 0)) = g−c u∗ + 1 − f (α × o(ϕ), ϕ , −c) + ug−c ∗ = 1 − (−g−c u + f (α × o(ϕ), ϕ , −c) − ug−c ). By the remark following Proposition 6.8, this is homotopic to 1 − e(α × o(ϕ), ϕ , −c), and moreover it is homotopic to 1 − e(α × o(ϕ), ϕ , 0) from Proposition 6.8. Thus, we get [θˆ (e)] = 1 − [e] in K0 (A Z2 ). In particular, θˆ is not approximately inner. Next, we would like to consider the map between K-groups induced from the inclusion ι : A → A Z2 . On the K0 -group, that is clearly given by ι∗0 ([f ]) = (0, [f ◦ π ]) ∈ Z ⊕ K 0 (X × Z2 , α × o(ϕ)) ∼ = K0 (A Z2 ) 0 ∼ for [f ] ∈ K (X, α) = K0 (A). On the K1 -group, for [f ] ∈ Coker(id −αϕ∗ ), we can see that ι∗1 ([f ]) = (0, [δ(f )]) ∈ Z ⊕ K 0 (X × Z2 , α × o(ϕ)) ∼ = K1 (A Z2 ), where δ(f )(x, k) = (−1)k f (x) for (x, k) ∈ X × Z2 . Under the identification Coker(id −α ∗ ) ∼ = K 0 (X × Z2 , α × o(ϕ))/K 0 (X, α) ϕ
established in the discussion before Lemma 2.5, this map is also described by ι∗1 ([f ] + K 0 (X, α)) = (0, [f − f γ ]) for f ∈ C(X × Z2 , Z). We now consider minimality and rigidity of α × ϕ.
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Lemma 8.3. Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuous map. Suppose that α ×ϕ is not orientation preserving. Then α ×ϕ is minimal if and only if α × o(ϕ) × ϕ is minimal. Proof. We follow the notation used in the discussion above. By Lemma 8.1, we may replace α × o(ϕ) × ϕ by α × o(ϕ) × ϕπ . Suppose that α × o(ϕ) × ϕπ is minimal. If E ⊂ X × T is a closed α × ϕ-invariant subset, then (π × id)−1 (E) is a closed α × o(ϕ) × ϕπ -invariant subset of X × Z2 × T. It follows that (π × id)−1 (E) is empty or the whole X × Z2 × T. Namely E is empty or the whole X × T. Let us prove the converse. Assume that α × ϕ is minimal. Let E ⊂ X × Z2 × T be a minimal subset of α × o(ϕ) × ϕπ . Since (π × id)(E) is a closed α × ϕ-invariant subset, it must be equal to X × T. If E = γ (E), then E = X × Z2 × T and we have nothing to do. Suppose that E ∩ γ (E) is empty and E ∪ γ (E) = X × Z2 × T. Thus E is a clopen subset. Hence there exists a continuous function χ : X → Z2 such that E = {(x, χ (x), t) : x ∈ X, t ∈ T}. It follows that χ + o(ϕ) = χ α, which contradicts [o(ϕ)] = 0 in K 0 (X, α)/2K 0 (X, α).
Lemma 8.4. Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuous map. Suppose that α × ϕ is not orientation preserving. (1) If α × o(ϕ) × ϕ is rigid, then α × ϕ is rigid. (2) If ϕ takes its values in Isom(T) and α × ϕ is rigid, then α × o(ϕ) × ϕ is rigid. Proof. We follow the notation used in the discussion above. By Lemma 8.1, we may ) denote the canonical factor replace α × o(ϕ) × ϕ by α × o(ϕ) × ϕπ. Let F (resp. F map from (X × T, α × ϕ) to (X, α) (resp. from (X × Z2 × T, α × o(ϕ) × ϕπ ) to (X × Z2 , α × o(ϕ)). (1) Suppose that there exist two distinct ergodic measures ν1 and ν2 for (X×T, α×ϕ) such that F∗ (ν1 ) = F∗ (ν2 ). Let ν˜ i be an α × o(ϕ) × ϕπ -invariant measure such that (π × id)∗ (˜νi ) = νi . Of course, ν˜ 1 = ν˜ 2 , because of (π × id)∗ (˜ν1 ) = ν1 = ν2 = (π × id)∗ (˜ν2 ). By replacing ν˜ i by 1 (˜νi + (γ × id)∗ (˜νi )), 2 ∗ (˜νi ) is invariant under γ . we may assume that ν˜ i is γ × id-invariant. It follows that F Together with ∗ (˜ν1 ) = F∗ (π × id)∗ (˜ν1 ) = F∗ (ν1 ) = F∗ (ν2 ) = F∗ (π × id)∗ (˜ν2 ) = π∗ F ∗ (˜ν2 ), π∗ F ∗ (˜ν1 ) = F ∗ (˜ν2 ). Therefore α × o(ϕ) × ϕπ is not rigid. we have F (2) Assume that α × o(ϕ) × ϕπ is not rigid. There exists an ergodic measure µ for ∗−1 (µ) is not a singleton. By assumption, ϕ takes its (X × Z2 , α × o(ϕ)) such that F values in the rotation group. It follows from Lemma 4.4 and its proof that there exist ∗−1 (µ). uncountably many ergodic measures for (X × Z2 × T, α × o(ϕ) × ϕπ ) in F
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∗−1 (µ) such that In particular, we can find two distinct ergodic measures ν1 , ν2 ∈ F (γ × id)∗ (ν1 ) = ν2 . Hence, it is easily verified that (π × id)∗ (ν1 ) = (π × id)∗ (ν2 ) in Mα×ϕ . But, we have ∗ (ν1 ) = π∗ (µ) = π∗ F ∗ (ν2 ) = F∗ (π × id)∗ (ν2 ), F∗ (π × id)∗ (ν1 ) = π∗ F and so α × ϕ is not rigid.
Now we consider cocycles with values in Isom(T). Let (X, α) be a Cantor minimal system and let ϕ : X → Isom(T) be a continuous map. There exists ξ ∈ C(X, T) such that ϕx = λo(ϕ)(x) Rξ(x) for all x ∈ X. Suppose that α × ϕ is not orientation preserving and not minimal. By Lemma 8.3, α × o(ϕ) × ϕ is not minimal, where ϕ is given by Rξ(x) k=0 ϕ(x,k) = λk+o(ϕ)(x) ϕx λk = λk Rξ(x) λk = R−ξ(x) k = 1. It is convenient to introduce δ : C(X, T) → C(X × Z2 , T) defined by δ(η)(x, k) = (−1)k η(x). It follows from Lemma 4.2 that there exist n ∈ N and ζ ∈ C(X × Z2 , Z) such that nδ(ξ ) = ζ − ζ ◦ (α × o(ϕ))−1 . We also have −nδ(ξ ) = nδ(ξ ) ◦ γ = (ζ − ζ ◦ (α × o(ϕ))−1 ) ◦ γ = ζ γ − ζ γ ◦ (α × o(ϕ))−1 . Since α × o(ϕ) is minimal on X × Z2 , ζ + ζ γ must equal a constant function. We can adjust ζ by a constant function so that ζ + ζ γ is equal to zero. Thus there exists η ∈ C(X, T) such that nδ(ξ ) = δ(η) − δ(η) ◦ (α × o(ϕ))−1 . Combining this with δ ◦ αϕ∗ = (α × o(ϕ))∗ ◦ δ, we obtain nξ = η − αϕ∗ (η). Lemma 8.5. Let (X, α) be a Cantor minimal system and let ϕx = λo(ϕ)(x) Rξ(x) be a cocycle with values in Isom(T). If α × ϕ is not orientation preserving and not minimal, then there exist n ∈ N and η ∈ C(X, T) such that nξ = η − αϕ∗ (η). Moreover, every minimal subset of α × ϕ is given by Es = {(x, t) : nt = αϕ∗ (η)(x) + s or nt = αϕ∗ (η)(x) − s} for some s ∈ T. Proof. The first part follows the discussion above. Let us consider the latter part. By Lemma 4.2, every minimal set of α × o(ϕ) × ϕ is given by {(x, k, t) : nt = (−1)k+o(ϕ)(α
−1 (x))
ηα −1 (x) + s}.
This closed set is carried to Es = {(x, t) : nt = αϕ∗ (η)(x) + s or nt = αϕ∗ (η)(x) − s} by the factor map from X × Z2 × T to X × T. Therefore Es is a minimal set of α × ϕ.
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9. Examples Example 9.1. Let θ ∈ T be an irrational number and let ξ : T → T be a continuous map. A homeomorphism γ : (s, t) → (s + θ, t + ξ(s)) on T2 is called a Furstenberg transformation. In [OP], the crossed product C ∗ -algebra arising from (T2 , γ ) is studied. We would like to replace the irrational rotation with a Cantor minimal system and construct an almost one to one extension of (T2 , γ ) as follows. Let ϕ be a Denjoy homeomorphism on T with r(ϕ) = θ as in Remark 3.2. The homeomorphism ϕ has the unique invariant nontrivial closed subset X. Let α be the restriction of ϕ to X. Then (X, α) is a Cantor minimal system and there exists an almost one to one factor map π from (X, α) to the irrational rotation (T, Rθ ) (see [PSS] for details). Both (X, α) and (T, Rθ ) are uniquely ergodic. It is easy to see that π × id : X × T → T2 satisfies (π × id) ◦ (α × Rξ π ) = γ ◦ (π × id), that is, π × id is a factor map. One can check that if γ is minimal, then α × Rξ π is also minimal. The factor map π induces a Borel isomorphism between X and S 1 . Hence α × Rξ π is rigid if and only if γ is uniquely ergodic. There are some known criteria for unique ergodicity of γ . For example, it was proved in [F, Theorem 2.1] that if ξ is a Lipschitz function and its degree is not zero then γ is uniquely ergodic. On the other hand it is known that there exist θ ∈ T and ξ : T → T such that γ is minimal but not uniquely ergodic (see [F, p.585] for instance). In this case α × Rξ π is minimal but not rigid. Now let us consider the example of Putnam which was presented by N. C. Phillips in [Ph3]. Example 9.2. For any θ ∈ R \ Q let gθ be a minimal homeomorphism of a Cantor set Xθ ⊂ T obtained from a Denjoy homeomorphism gθ0 : T → T as in [Ph3]. Choose gθ0 to have rotation number θ and such that the unique minimal set Xθ ⊂ T has the property that the image of T \ Xθ under the semiconjugation to Rθ is a single orbit of Rθ . Now let θ1 , θ2 ∈ R \ Q be irrational numbers such that 1, θ1 , θ2 are Q-linearly independent. Consider two systems (Xθ1 × T, gθ1 × Rθ2 ) and (Xθ2 × T, gθ2 × Rθ1 ). Then both are minimal. Let A = C ∗ (Xθ1 × T, gθ1 × Rθ2 ) and B = C ∗ (Xθ2 × T, gθ2 × Rθ1 ). It follows from [Ph3, Proposition 1.12] that K1 (A) ∼ = K1 (B) = Z3 and (K0 (A), K0 (A)+ , [1A ]) ∼ = (Z + θ1 Z + θ2 Z, (Z + θ1 Z + θ2 Z)+ , 1) ∼ = (K0 (B), K0 (B)+ , [1B ]),
where Z + θ1 Z + θ2 Z ⊂ R. In particular, both systems are rigid. Moreover A and B have tracial rank zero and they are isomorphic by the classification theorem in [L5]. However, there is no order isomorphism between Z + θ1 Z and Z + θ2 Z. It follows from Theorem 7.9 that they are not approximately K-conjugate. On the other hand, by [M3, Corollary 4.10], they are weakly approximately conjugate. When we choose another continuous function ξ : Xθ1 → T, two systems gθ1 × Rθ2 and gθ1 × Rξ may not be conjugate. But, if the integral value of ξ is equal to θ2 , then we can conclude that they are approximately K-conjugate by Theorem 7.8.
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Example 9.3. We would like to construct a non-orientation preserving minimal homeomorphism on X × T concretely. Let (X, α) be an odometer system of type 3∞ . It is well-known that K 0 (X, α) is isomorphic to Z[1/3]. We regard X as a projective limit of Z3n and denote the canonical projection from X to Z3n by πn . Notice that πn (α(x)) = πn (x) + 1 for all x ∈ X, where the addition is understood modulo 3n . Let x0 be the point of X such that πn (x0 ) = 0 for all n ∈ N. We will construct a continuous map ϕ : X → Isom(T) of the form ϕx = λRξ(x) so that α × ϕ is a minimal homeomorphism on X × T. Note that o(ϕ)(x) = 1 and [o(ϕ)] = 0 in K 0 (X, α)/2K 0 (X, α) ∼ = Z2 . By Lemma 8.5, if the closure of {(α × ϕ)m (x0 , 0) : m ∈ N} contains {x0 } × T, then we can deduce the minimality of α × ϕ. Let {tn }n∈N be a dense n sequence of T. Since α 3 (x0 ) → x0 as n → ∞, it suffices to construct ϕ so that n n (α × ϕ)3 (x0 , 0) = (α 3 (x0 ), −tn ). Let sn ∈ (−2−1 , 2−1 ] be the real number satisfying sn + Z = tn − tn−1 , where we put t0 = 0. We define a map ξn : X → T by 0 πn (x) = 0, 1, . . . , 3n−1 − 1 ξn (x) = (−1)k s + Z otherwise. 2·3n−1 n For all n, m ∈ N, it is not hard to see that n −1 3
0 n<m sm + Z n ≥ m.
(−1)k ξm (α k (x0 )) =
k=0
Since |ξn (x)| <
3−n
for every x ∈ X, ξ(x) =
∞
ξn (x)
n=1
exists and is continuous on X. Put ϕx = λRξ(x) for all x ∈ X. Then n −1 3
(−1)k ξ(α k (x0 )) =
k=0
n
si + Z = t n
i=1
implies n
n
n
(α × ϕ)3 (x0 , 0) = (α 3 (x0 ), λ(tn )) = (α 3 (x0 ), −tn ) for all n ∈ N. It follows that α × ϕ is minimal. Example 9.4. We would like to construct a cocycle with values in Homeo+ (T) which is not cohomologous to a cocycle with values in the rotation group. It is useful to introduce a complete metric d(·, ·) of Homeo+ (T) defined by d(ϕ, ψ) = max |ϕ(t) − ψ(t)|, |ϕ −1 (t) − ψ −1 (t)| . t∈T
In the argument below, we use the following facts.
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Fact (a). For any s, t ∈ T and ε > 0, there exists ρ ∈ Homeo+ (T) such that |ρ(s)−s| < ε, |ρ(t) − s| < ε and ρ is conjugate to an irrational rotation. Fact (b). Homeo+ (T) is arcwise connected. Let us construct two sequences of natural numbers mn and ln , and a sequence of maps ϕn : Zmn → Homeo+ (T) inductively so that the following conditions are satisfied. It is convenient to view ϕn as a periodic map from Z. (1) mn−1 divides mn and ln is not greater than mn /mn−1 . (2) ψn = ϕn (mn − 1) . . . ϕn (1)ϕn (0) is conjugate to an irrational rotation. (3) Both |ψn (0)| and |ψn (1/2)| are less than 1/n. k (t) : k = 1, 2, . . . , l } is 1/n-dense in T. (4) For every t ∈ T, {ψn−1 n (5) For every k = 0, 1, . . . , ln mn−1 − 1, ϕn (k) = ϕn−1 (k). (6) For every k ∈ Zmn , d(ϕn (k), ϕn−1 (k)) is less than 2−n . If these conditions are achieved, then we can finish the proof as follows. Let (X, α) be the odometer system of type {mn }n . Namely, X is the projective limit of Zmn and there exists a natural projection πn : X → Zmn such that πn (α(x)) = πn (x) + 1, where the addition is understood modulo mn . Let x0 be the point of X such that πn (x0 ) = 0 for all n ∈ N. From (6), ϕx = lim ϕn (πn (x)) ∈ Homeo+ (T) n→∞
exists for all x ∈ X and ϕ is a continuous map from X to Homeo+ (T). By (4) and (5), for all n ∈ N and t ∈ T, we can see that {ϕα k−1 (x0 ) . . . ϕα(x0 ) ϕx0 (t) : k = mn−1 , 2mn−1 , . . . , ln mn−1 } is 1/n-dense in T. Hence, for every t ∈ T, the closure of {(α × ϕ)k (x0 , t) : k ∈ N} in X × T contains {x0 } × T. It follows that α × ϕ is minimal. Let us check that ϕ is never cohomologous to a cocycle with values in the rotation group. Suppose that ϕ is cohomologous to a cocycle with values in the rotation group. Then we would have a homeomorphism id ×γ on X × T such that (id ×γ )(α × ϕ) = (α × ϕ)(id ×γ ) and (id ×γ )(x0 , 0) = (x0 , 1/2). By (3), we can verify that (α × ϕ)mn (x0 , 0) → (x0 , 0) and (α × ϕ)mn (x0 , 1/2) → (x0 , 0) in X × T as n → ∞. Consequently we obtain (id ×γ )(x0 , 0) = lim (id ×γ )(α × ϕ)mn (x0 , 0) n→∞
= lim (α × ϕ)mn (id ×γ )(x0 , 0) n→∞
= lim (α × ϕ)mn (x0 , 1/2) = (x0 , 0), n→∞
which is a contradiction.
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Let us construct mn , ln and ϕn . Put m1 = l1 = 1 and let ϕ1 be an irrational rotation. Suppose that mn−1 , kn−1 and ϕn−1 have been fixed. Put ψn−1 = ϕn−1 (mn−1 − 1) . . . ϕn−1 (1)ϕn−1 (0). By (2), there exists ω ∈ Homeo+ (T) such that ωψn−1 ω−1 = Rθ , where r(ψn−1 ) = θ is an irrational number. Since ψn−1 is minimal, we can find a natural number ln so that the condition (4) holds. Applying Fact (a) to ω(0) and ω(1/2), we obtain ρ ∈ Homeo+ (T) such that both |ω−1 ρω(0)| and |ω−1 ρω(1/2)| are less than 1/n. Furthermore ρ is conjugate to an irrational rotation. Define ω˜ : Zmn−1 → Homeo+ (T) by ω(k) ˜ = ωϕn−1 (0)−1 ϕn−1 (1)−1 . . . ϕn−1 (k − 1)−1 for all k = 0, 1, . . . , mn−1 − 1. Evidently we have id k = mn−1 − 1 −1 ω(k ˜ + 1)ϕn−1 (k)ω(k) ˜ = Rθ k = mn−1 − 1 for all k ∈ Zmn−1 . Choose ε > 0 so that d(ϕ, id) < ε implies 1 2n for all k ∈ Zmn−1 . By Fact (b), we can find a natural number N greater than ln and a sequence of homeomorphisms d(ϕn−1 (k)ω(k) ˜ −1 ϕ ω(k), ˜ ϕn−1 (k)) <
id = ρ0 , ρ1 , . . . , ρmn−1 (N−ln ) = ρRθ−N such that d(ρi+1 ρi−1 , id) < ε for all i = 0, 1, . . . , mn−1 (N − ln ) − 1. Note that this is easily done because d(·, ·) is invariant under rotations. Put mn = mn−1 N . We define a map ρ˜ : Zmn → Homeo+ (T) by id k = 0, 1, . . . , ln mn−1 − 1 ρ(k) ˜ = −j j otherwise, Rθ ρk +1 ρk−1 Rθ where k = k − ln mn−1 and j is a natural number satisfying N − k/mn−1 ≤ j < N + 1 − k/mn−1 . Notice that we still have d(ρ(k), ˜ id) < ε for all k ∈ Zmn . Define ϕn : Zmn → Homeo+ (T) by ϕn (k) = ϕn−1 (k)ω(k) ˜ −1 ρ(k) ˜ ω(k). ˜ The condition (5) is already built in this definition. The condition (6) is immediate from the choice of ε. Since one can check that ψn = ϕn (mn − 1) . . . ϕn (1)ϕn (0) = ω(m ˜ n )−1 (Rθ ρ(m ˜ n − 1) . . . ρ(m ˜ n − mn−1 )Rθ . . . ρ(l ˜ n mn−1 )) Rθln ω(0) ˜ = ω−1 ρRθ−N RθN ω = ω−1 ρω, the conditions (2) and (3) follow immediately.
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Acknowledgement. The first named author would like to acknowledge the support from a NSF grant. He would also like to thank the second author for his effort to make this project possible. He was also partially supported by Shanghai Academic Priority Disciplines. The second named author was supported by Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science. He is grateful to Yoshimichi Ueda and Hiroyuki Osaka for helpful advice.
References [B]
Blackadar, B.: K-theory for operator algebras. Mathematical Sciences Research Institute Publications 5. Cambridge: Cambridge University Press, 1998 [BBEK] Blackadar, B., Bratteli, O., Elliott, G. A., Kumjian, A.: Reduction of real rank in inductive limits of C ∗ -algebras. Math. Ann. 292, 111–126 (1992) [BKR] Blackadar, B., Kumjian, A., Rørdam, M.: Approximately central matrix units and the structure of noncommutative tori. K-Theory 6, 267–284 (1992) [EL] Exel, R., Loring, T. A.: Invariants of almost commuting unitaries. J. Funct. Anal. 95, 364–376 (1991) [F] Furstenberg, H.: Strict ergodicity and transformation of the torus. Amer. J. Math. 83, 573–601 (1961) [GPS] Giordano, T., Putnam, I. F., Skau, C. F.: Topological orbit equivalence and C ∗ -crossed products. J. reine angew. Math. 469, 51–111 (1995) [GW] Glasner, E., Weiss, B.: Weak orbit equivalence of Cantor minimal systems. Internat. J. Math. 6, 559–579 (1995) [H] Haagerup, U.: Every quasitrace on an exact C ∗ -algebra is a trace. Preprint, 1991 [HPS] Herman, R. H., Putnam, I. F., Skau, C. F.: Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3, 827–864 (1992) [HLX] Hu, S., Lin, H., Xue, Y.: The tracial topological rank of extensions of C ∗ -algebras. Math. Scand. 94, 125–147 (2004) [KH] Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications, 54. Cambridge: Cambridge University Press, 1995 [K] Kishimoto, A.: Automorphisms of AT algebras with the Rohlin property. J. Operator Theory 40, 277–294 (1998) [KK] Kishimoto, A., Kumjian, A.: Crossed products of Cuntz algebras by quasi-free automorphisms. In: Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun. 13, Providence, RI: Amer. Math. Soc., 1997, pp. 173–192 [L1] Lin, H.: Tracially AF C ∗ -algebras. Trans. Amer. Math. Soc. 353, 693–722 (2001) [L2] Lin, H.: The tracial topological rank of C ∗ -algebras. Proc. London Math. Soc. 83, 199–234 (2001) [L3] Lin, H.: An introduction to the classification of amenable C ∗ -algebras. River Edge, NJ: World Scientific Publishing Co., Inc., 2001 [L4] Lin, H.: Classification of simple C ∗ -algebras and higher dimensional noncommutative tori. Ann. of Math. (2) 157, 521–544 (2003) [L5] Lin, H.: Classification of simple C ∗ -algebras with tracial topological rank zero. Duke Math. J.125, N0. 1, 91–119 (2004) [L6] Lin, H.: Classification of homomorphisms and dynamical systems. http://arxiv.org/ list/math.OA/0404018, 2004. Trans. Amer. Math. Soc., to appear [LM] Lin, H., Matui, H.: Minimal dynamical systems and approximate conjugacy. http:// arxiv.org/list/math.OA/0402309, 2004 [LO] Lin, H., Osaka, H.: The Rokhlin property and the tracial topological rank. J. Funct. Anal. 218, 475–494 (2005) [LP1] Lin, Q., Phillips, N. C.: Direct limit decomposition for C ∗ -algebras of minimal diffeomorphisms. In: The Proceedings of the US-Japan Operator Algebra Conference in Fukuoka, Poerator Algebra and Applications, Adv. Studies in Pure Mathematics, Vol. 38, Tokyo: Math. Soc. of Japan, 2004, pp. 107–133 [LP2] Lin, Q., Phillips, N. C.: The structure of C ∗ -algebras of minimal diffeomorphisms. In preparation [M1] Matui, H.: Ext and OrderExt classes of certain automorphisms of C ∗ -algebras arising from Cantor minimal systems. Canad. J. Math. 53, 325–354 (2001) [M2] Matui, H.: Finite order automorphisms and dimension groups of Cantor minimal systems. J. Math. Soc. Japan 54, 135–160 (2002)
Minimal Dynamical Systems on the Product of the Cantor Set and the Circle [M3] [OP] [Pa] [Ph1] [Ph2] [Ph3] [Pu1] [Pu2] [PSS] [R] [T] [W] [Z]
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Matui, H.: Approximate conjugacy and full groups of Cantor minimal systems. http://arxiv.org/list/math.DS/0404224, 2004 Osaka, H., Phillips, N. C.: Furstenberg transformations on irrational rotation algebras. Preprint Parry, W.: Compact abelian group extensions of discrete dynamical systems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13, 95–113 (1969) Phillips, N. C.: Cancellation and stable rank for direct limits of recursive subhomogeneous algebras. http://arxiv.org/list/ math.OA/0101157, 2001 Phillips, N. C.: Crossed products of the Cantor set by free minimal actions of Zd . http://arxiv.org/list/math.OA/0208085, 2002 Phillips, N. C.: When are crossed products by minimal diffeomorphisms isomorphic?. Operator algebras and mathematical physics (Constant¸a, 2001), Bucharest: In: Theta, 2003, pp. 341–364 Putnam, I. F.: The C ∗ -algebras associated with minimal homeomorphisms of the Cantor set. Pacific J. Math. 136, 329–353 (1989) Putnam, I. F.: On the K-theory of C ∗ -algebras of principal groupoids. Rocky Mountain J. Math. 28 , 1483–1518 (1998) Putnam, I. F., Schmidt, K., Skau, C. F.: C ∗ -algebras associated with Denjoy homeomorphisms of the circle. J. Operator Theory 16, 99–126 (1986) Renault, J.: A groupoid approach to C ∗ -algebras. Lecture Notes in Mathematics 793, Berlin: Springer 1980 Tomiyama, J.: Topological full groups and structure of normalizers in transformation group C ∗ -algebras. Pacific J. Math. 173, 571–583 (1996) Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79, New YorkBerlin: Springer- Verlag, 1982 Zimmer, R. J.: Extensions of ergodic group actions. Illinois J. Math. 20, 373–409 (1976)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 257, 473–497 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1328-3
Communications in
Mathematical Physics
Combining Multifractal Additive and Multiplicative Chaos Julien Barral, St´ephane Seuret INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. E-mail: [email protected]; [email protected] Received: 2 June 2004 / Accepted: 21 October 2004 Published online: 12 April 2005 – © Springer-Verlag 2005
Abstract: The purpose of this article is the study of the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ ,σ =
b−j γ j ≥1
j2
µ([kb−j , (k + 1)b−j ))σ δkb−j (γ ≥ 0, σ ≥ 1),
0≤ k ≤bj −1
where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities analysis of the measures νγ ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum dνγ ,σ of νγ ,σ is linear on [0, hγ ,σ ] for some critical value hγ ,σ . Then dνγ ,σ is strictly concave on the right of hγ ,σ , and on this part it is deduced from the multifractal spectrum of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. They open interesting perspectives in modeling discontinuous phenomena. 1. Introduction The multifractal nature of functions or measures possessing jump discontinuities has been investigated in several situations [30, 51, 31, 22, 23]. The purpose of this article is the construction and the multifractal analysis of a new class of measures defined by infinite sums of Dirac masses. The study of these measures gives rise to yet unknown multifractal behaviors. Moreover, this class illustrates most of the multifractal behaviors one can expect from discontinuous measures which satisfy some multifractal formalism. This is important for the purpose of modeling discontinuous phenomena which are known to exhibit multifractal behaviors. Such behaviors occur for instance in geophysics [28] when considering the spatial-temporal position and the intensity of seismic events,
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in telecommunications where the TCP Internet traffic is known to be multifractal [38], and also when studying financial time series [41]. The local regularity of a function or a measure µ at a point x is usually described by an H¨older exponent hµ (x). Our work draws its interest from positive Borel measures, and in this case the H¨older exponent is defined by hµ (x) = lim inf r→0+
log µ(B(x, r)) , log r
where B(x, r) stands for the closed ball of radius r centered at x. The multifractal analysis of µ consists in computing the size of the level sets of this µ H¨older exponent h, Eh = {x : hµ (x) = h}. More precisely, one often tries to find the Hausdorff multifractal spectrum dµ of the measure µ defined by µ
h → dµ (h) = dim(Eh ), where dim E stands for the Hausdorff dimension of the set E. Multifractal analysis started in the context of the study of fully developed turbulence with the following heuristics: In [26], Frisch and Parisi proposed a connexion, via a Legendre transform, between the Hausdorff multifractal spectrum of the energy dissipation measure µ and a kind of free energy function associated with µ. In the recent past years, a substantial amount of work has been devoted to compute the multifractal spectra of several classes of functions and measures [27, 16, 50, 15, 29, 20, 44, 1, 43, 48, 37, 47, 54, 5, 7, 25]. These studies confirmed this connexion, which is now known as multifractal formalism. We make precise the definition of this formalism in a moment. Among the measures which multifractal analysis has been performed, two families can be distinguished by the typical shape of their spectrum. Some measures, the construction of which is based on an additive scheme, exhibit linear increasing spectrum (see Fig. 1): There exists β ∈ (0, 1] such that dµ (h) = βh for 0 ≤ h ≤ 1/β. L´evy subordinators [31] and the sums of Dirac masses of [22] belong to this class. These measures are a form of additive chaos. In these specific cases, the H¨older exponent at each point x is closely connected to the approximation rate of x by jump points as well as to the masses carried by these points. In this framework, the notion of “ubiquity” of some “resonant” sets [2, 18, 19] is accountable for the linear shape of the multifractal spectrum. Atomless measures with a construction involving a multiplicative scheme usually have a strictly concave spectrum, including a decreasing part (see Fig. 1). Multinomial measures, quasi-Bernoulli measures, Mandelbrot cascades and their extensions, as well as the recent compound Poisson cascades, are examples of such multiplicative chaos measures [14, 40, 33, 17, 7, 8]. These measures typically have a multifractal spectrum with the well-known ∩-shape, reflecting the validity of a multifractal formalism. This follows from the Large Deviations theory (or from a similar argument) applied to the elements of a family of auxiliary “Gibbs” measures {µh }h≥0 such that each µh is carried µ by the level set Eh . It is natural to try to mix these two distinct construction schemes. In this article, we put the following scheme forward, where the jump points are the b-adic points. The heterogeneity in the distribution of the masses assigned to these points is created with the use of an auxiliary measure µ. More precisely, if µ is a positive Borel measure on [0, 1], let us consider the measure νγ ,σ defined with the help of two parameters γ ≥ 0 and σ ≥ 1 by
Multifractal Additive and Multiplicative Chaos
νγ ,σ =
j ≥1
j −2
475
b−j γ µ([kb−j , (k + 1)b−j ))σ δkb−j .
(1)
0≤k≤bj −1
−2 The factor j −2 makes the series converge when γ = 0. In fact {j }j ≥1 could be replaced by any decreasing positive sequence {aj }j ≥1 such that j ≥1 aj < +∞ and | log aj | = o(j ). This class of measures has a fruitful structure, and it provides new important examples of measures that fulfill a multifractal formalism. Moreover, the measures νγ ,σ have their natural counterparts in terms of discontinuous function series and wavelet series (see [12, 10]). Let us mention that sets other than b-adic numbers could have been chosen for the location of the Dirac masses. Similar constructions will be performed in further works, using the rational numbers or some random families of points, as well as suitable associated weights. The construction we deal with in this paper is key to understand the main ideas that rule the mixing between additive and multiplicative chaos. For the sake of comprehensibility, we also choose to work in the one-dimensional case. In order to fully understand the next results, let us now resume the notion of multifractal formalism. A multifractal formalism for measures relates the multifractal spectrum dµ to the Legendre transform of a scaling function associated with µ (see [14, 45] for complete mathematical foundations).A possible definition for the scaling function [14] is q 1 τµ : q ∈ R → τµ (q) = lim inf − logb µ [kb−j , (k + 1)b−j ) , (2) j →+∞ j j 0≤k≤b −1
= 0 ∀q (τµ does not depend on b if supp(µ) = [0, 1]). with the convention In this paper, the multifractal formalism is said to hold for µ at exponent h when the multifractal spectrum coincides with the Legendre transform of the scaling function at h, µ i.e. when dim Eh = dµ (h) = τµ∗ (h) := inf q∈R (qh − τµ (h)). This formalism combines some level sets considered in [45] and the scaling function of [14], and is satisfied by the classes of measures mentioned above. Moreover, if one defines 0q
qc (µ) = inf{q : τµ (q) = 0} and hc (µ) = τµ (qc (µ)− ),
(3)
a linear spectrum starting at (0, 0) is equivalent to the fact that hc (µ) > 0 and τµ (qc (µ)+ ) = 0. Eventually, the spectrum exhibits a concave part on the right side of τµ (qc (µ)− ) as soon as τµ is not linear when q < qc (µ). Notice that one always has 0 < qc (µ) ≤ 1 and 0 ≤ hc (µ) ≤ qc (µ)−1 . We describe the properties and the multifractal structure of νγ ,σ in two steps. It is convenient to begin with the basic construction ν = ν0,1 , and then to look at the influence of the parameters (γ , σ ). In order to state our results, three technical conditions detailed along this paper are required: Condition C1 ensures that the µ-mass of the b-adic intervals do not converge to 0 too fast as the intervals lengths converge to 0. C2(h) requires that µ possesses some statistical self-similarity property and that there exists a control of the “speed of renewal” of the level sets of the H¨older exponents of µ (see properties (3) and (4) in Sect. 3.1). C3(h) is weaker than C2(h) and implies the validity of the multifractal formalism for µ at h. Though technical, conditions C1-3 are rather natural and are satisfied by many classes of measures, as for instance the statistically self-similar measures µ mentioned above obtained as limits of multiplicative processes. Examples of such measures are detailed in Sect. 3.2.
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1
o
1
ht
h
o
1
ht
h
o
hc
ht
h
Fig. 1. Typical multifractal spectrum of Left: a measure µ built on an additive scheme, Middle: on a multiplicative scheme, Right: of a measure ν under suitable assumptions on µ. Here ht is the Lebesgue-almost sure exponent
Theorem 1. Let µ be a positive Borel measure such that supp(µ) = [0, 1], and assume that C1 holds for µ. Let ν = ν0,1 be the measure given by formula (1). 1. If hc (µ) > 0, for every h ∈ [0, hc (µ)] one has dν (h) ≤ qc (µ)h. If C2(hc (µ)) holds, for every h ∈ [0, hc (µ)] one has dν (h) = qc (µ)h, and the multifractal formalism holds at h. Moreover, qc (ν) = qc (µ) and hc (ν) = hc (µ). 2. Let h ≥ hc (µ). Then dν (h) ≤ τµ∗ (h) if τµ∗ (h) ≥ 0, and Ehν = ∅ if τµ∗ (h) < 0. If C3(h) holds, then dµ (h) = dν (h) = τµ∗ (h) = τν∗ (h), and the multifractal formalism holds at h. Theorem 1 applies to the measure ν itself: the process can be iterated, the spectrum being unchanged. We shall see that τν (q) ≤ τµ (q) if q ≤ qc (µ) and τµ∗ (τµ (q + )) ≥ 0, and that τν (q) = 0 if q > qc (µ). There is equality everywhere when C3(τµ (q + )) holds for a dense countable set of q’s such that τµ∗ (τµ (q + )) ≥ 0 and τµ (q + ) ≥ hc (µ). When hc (µ) > 0, it is tempting, by analogy with the thermodynamical frame, to think about the non-differentiability of τν at qc (ν) as a phase transition (see [53] and also [25] for discussions). It might be of interest to establish whether there is a link between our construction and this sort of phenomenon. The following remark is key. Under the assumptions of Theorem 1 and when hc (µ) = ν = {x : τµ (qc (µ)− ) > 0, multifractal formalisms that focus on level sets such as E h = h} (defined using a limit rather than a lim inf) do not hold for ν at limr→0 log ν(B(x,r)) log r h when 0 < h < hc (µ). This was noticed in [3] where the authors consider the measure νγ ,1 in the case where µ is the Lebesgue measure. The same difficulty is encountered in [51] with some self-similar sums of Dirac masses (close to our class ν0,1 when µ is multinomial). Nevertheless [51] concludes to a failure of the multifractal formalism ν were considered. since only the sets E h This phenomenon pleads for the choice of the sets Ehν defined using a lim inf, because no information is lost: These sets always form a partition of [0, 1]. This choice led us to investigate in detail the repartition of the mass of µ. More precisely, the validity of Theorem 1 depends on the following theorem, which gives a lower bound for the dimension of sets that are related to µ and to some approximation rate by b-adic numbers. If ψ is a continuous positive function with ψ(0) = 0, and if h > 0, then Qhψ (I ) is said to hold for an interval I when |I |h+ψ(|I |) ≤ µ(I ) ≤ |I |h−ψ(|I |) .
Theorem 2. Let µ be a positive Borel measure such that supp(µ) = [0, 1], and h > 0. For every ξ > 1, for every continuous positive function ψ with ψ(0) = 0 and for every positive sequence ε = {εj }j ≥1 converging to 0, let us define
Multifractal Additive and Multiplicative Chaos
Sξ,ε,ψ (h) =
477
n≥1 j ≥n
k∈{0,... ,bj −1}:
[kb−j , kb−j + b−j (ξ −εj ) ].
(4)
Qhψ ([kb−j ,(k+1)b−j )) holds
Suppose that C2(h) holds. There exists a function ψ such that for every ξ > 1, one can find a positive sequence ε converging to 0 and a positive Borel measure mξ on [0, 1] with the following properties: mξ (Sξ,ε,ψ (h)) > 0, and for every Borel set E ⊂ [0, 1] with dim E < τµ∗ (h)/ξ , mξ (E) = 0. Thus, dim Sξ,ε,ψ (h) ≥ τµ∗ (h)/ξ . Theorem 2 appears to be the consequence of a stronger result, Theorem 3, that we establish in Sect. 3. Theorem 2 and 3 apply to the measures µ mentioned above as illustrations of Theorem 1 and described in Sect. 3.2. Let us recall that if x ∈ R, and ξ ≥ 1, x is said to be ξ -approximated if there exist an infinite number of b-adic numbers kb−j such that |kb−j − x| ≤ b−j ξ . With each x is associated its approximation rate ξx = sup{ξ ≥ 1 : x is ξ -approximated}.
(5)
One always has ξx ≥ 1, and it is shown in [20, 32] for example that the set {x ∈ R : ξx = ξ } has a Hausdorff dimension equal to 1/ξ . Theorem 2 allows the computation of the Hausdorff dimension of the set of points that are infinitely often close at rate ξ to b-adic numbers kb−j that verify µ([kb−j , (k + 1)b−j )) ∼ b−j h . Theorems 2 and 3 are referred to as “measure-conditioned ubiquity”. They yield a generalization of the notion of ubiquity (see [19]), in the sense that they involve an ubiquity property (i.e. an omnipresence) of sets of points that must satisfy some property. Here we work with b-adic points in [0, 1] and the property is related to the behavior of µ [kb−j , (k + 1)b−j ) . In our context, the “usual” ubiquity theorems [18, 19, 32] shall be understood as Theorem 3 applied to µ = λ (the Lebesgue measure), and in this case Qhψ (I ) corresponds to a trivial condition. The property of the lim sup-sets Sξ,ε,ψ (h) to be non-empty is thus remarkable, and strongly depends on the measure µ considered. Let us now consider the measures νγ ,σ defined by (1), where γ ≥ 0, σ ≥ 1. Theorem 1’ . Let µ be a positive Borel measure such that supp(µ) = [0, 1], and assume that C1 holds for µ. Let γ ≥ 0 and σ ≥ 1. Let qγ ,σ = inf{q ∈ R : τµ (σ q) + γ q = 0}, and hγ ,σ = σ τµ (σ qγ−,σ ) + γ . 1. If hγ ,σ > 0, for every h ∈ [0, hγ ,σ ], dνγ ,σ (h) ≤ qγ ,σ h. h
−γ
If C2( γ ,σσ ) holds, for every h ∈ [0, hγ ,σ ], dνγ ,σ (h) = qγ ,σ h, and the multifractal formalism holds at h. Moreover, qγ ,σ = qc (νγ ,σ ) and hγ ,σ = hc (νγ ,σ ). ν 2. Let h ≥ hγ ,σ . Then dνγ ,σ (h) ≤ τµ∗ h−γ if τµ∗ h−γ ≥ 0, and Ehγ ,σ = ∅ if σ σ ∗ h−γ < 0. If C3( h−γ τµ∗ h−γ σ σ ) holds, then dνγ ,σ (h) = τµ σ , and the multifractal formalism holds at h. The spectrum dνγ ,σ has in fact the same shape as the one of dν (i.e. composed of two parts), but γ and σ allow us to “play” with the slope of the linear part and the shape of the (strictly) concave part. The measures νγ ,σ give the possibility to reach examples of measures m which illustrate all possible pairs 0 < qc (m) ≤ 1, 0 ≤ hc (m) ≤ qc−1 (m). This makes this class valuable. Until now, the case qc (m) < 1 and hc (m) > 0 was obtained only when
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qc (m)hc (m) = 1 and when m is the derivative of a L´evy subordinator [31] or m = νγ ,1 in the case where µ is the Lebesgue measure [3]. The cases qc (m) = 1, 0 < hc (m) ≤ 1 are reached for example with m = ν by using multinomial measures µ in Theorem 1. The introduction of the parameters γ and σ allows us to reach all the possibilities qc (m) < 1 and hc (m) > 0 with m = νγ ,σ and the same choice for µ. The case hc (µ) = 0 is particularly remarkable. In this case, C2(hc (µ)) of Theorem 1 is useless. When C3(h) is satisfied by µ for every h such that τµ∗ (h) > 0, dµ has the classical ∩-shape, and it begins at (0, 0). To our knowledge, this kind of behavior appears only in the case qc (µ) = 1 in [42, 51, 5]. The construction of m = ν0,σ with such measures illustrates the cases qc (m) < 1 and hc (m) = 0. Section 2 recalls the definitions of H¨older exponents and of the multifractal formalism adapted to our construction. Conditions C1 and C3(h) are given. Section 3 holds the definition of Condition C2(h) and the proof of Theorem 3, which implies Theorem 2. Subsection 3.2 indicates classes of measures µ that fulfill conditions C1-3, and thus yield explicit examples of measures ν. Section 4 contains the proof of Theorem 1. Some observations, especially concerning the validity of the multifractal formalism for νγ ,σ , are gathered in Sect. 5. 2. General Settings Fix b an integer greater than 2. For j ≥ 1 and k ∈ [0, . . . , bj − 1], one sets Ij,k = + − [kb−j , (k + 1)b−j ). Ij,k and Ij,k denote the intervals Ij,k + b−j and Ij,k − b−j . If x ∈ (0, 1), ∀j ≥ 1 Ij (x) denotes the b-adic interval of length b−j that contains x. Then define Ij+ (x) = Ij (x) + b−j and Ij− (x) = Ij (x) − b−j . For each j ≥ 1, kj,x is the unique integer such that Ij (x) = [kj,x b−j , (kj,x + 1)b−j ). A b-adic number kb−j is said to be irreducible if the fraction bkj is irreducible. |B| always denotes the diameter of the set B. Eventually, for the rest of the paper, we adopt the convention log(0) = −∞. 2.1. Local regularity of measures. Definition 1. Let µ be a positive Borel measure on [0, 1], x0 ∈ [0, 1]. One sets log µ(B(x0 , r)) log µ(B(x0 , b−j )) = lim inf . r→0+ log |B(x0 , r)| j →+∞ log |B(x0 , b−j )| The lower and upper H¨older exponents of µ at x0 are respectively defined by log µ(Ij (x0 )) log µ(Ij (x0 )) and α µ (x0 ) = lim sup . α µ (x0 ) = lim inf j →+∞ log |Ij (x0 )| j →+∞ log |Ij (x0 )| hµ (x0 ) = lim inf
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When α µ (x0 ) = α µ (x0 ), their common value is denoted αµ (x0 ) and called the H¨older exponent of µ at x0 . The left and right lower and upper H¨older exponents of µ at x0 are defined by α− µ (x0 ) = lim inf
j →+∞
and α − µ (x0 ) = lim sup j →+∞
log µ(Ij− (x0 )) log |Ij− (x0 )| log µ(Ij− (x0 )) log |Ij− (x0 )|
and α + µ (x0 ) = lim inf
j →+∞
and α + µ (x0 ) = lim sup j →+∞
log µ(Ij+ (x0 )) log |Ij+ (x0 )| log µ(Ij+ (x0 )) log |Ij+ (x0 )|
.
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Similarly, when they coincide, αµ− (x0 ) and αµ+ (x0 ) denote their common value. + The reader can check that hµ (x) = min(α − µ (x), α µ (x), α µ (x)).
Definition 2. For every positive Borel α ≥ 0, let measure µ on [0, 1] and for every µ αµ = x : αµ (x) = αµ+ (x) = αµ− (x) = α . Eα = {x : hµ (x) = α} and E µ The mapping dµ : α ≥ 0 → dim(Eα ) is called the multifractal spectrum of µ. One µ α . also sets d˜µ (α) = dim E 2.2. Legendre and Large Deviation spectrum, multifractal formalism. The Legendre transform of a function ϕ : R → R ∪ {−∞} is defined by ϕ ∗ : h → inf (ph − ϕ(p)). p∈R
Let µ be a positive Borel measure on [0, 1]. The function τµ defined by (2) is known to be concave, non-decreasing, and the mapping h → τµ∗ (h) is referred to as the Legendre spectrum of µ. Definition 3. Let µ be a positive Borel measure on R. Let us define, ∀α ≥ 0, η > 0 and log µ(Ij,k ) ∈ [α−η, α+η] , and dηg (α) = lim sup j −1 logb Nj,η (α). j ≥ 1, Nj,η (α) = # k : log b−j g
j →+∞
g
The large deviation spectrum of µ is the mapping dµ : α → limη→0+ dη (α). The following lemma follows from standard arguments. It gives a heuristic interpretation of the large deviation spectrum and is used in Sect. 4.3. Lemma 1. Let µ be a positive Borel measure on [0, 1]. For every 0 ≤ β ≤ α, for every ε > 0 and η > 0, there exists a scale J such that j ≥ J implies log # k : b−j α ≤ µ(Ij,k ) ≤ b−jβ ≤ sup dµg (α ) + ε. log bj
max(β−η,0)≤α ≤α+η The Legendre and large deviation spectra are useful in multifractal analysis, more on these topics can be found in [52]. They are more tractable than dµ , and they yield upper bounds for dµ . Remark that the maximum of α → τµ∗ (α) is always reached at τµ (0+ ). g Proposition 1. 1. Let α ≥ 0. One has d˜µ (α) ≤ dµ (α) ≤ τµ∗ (α) and d˜µ (α) ≤ dµ (α) ≤ µ τµ∗ (α). If τµ∗ (α) < 0 then Eα = ∅.
µ
+ 2. If α ∈ [0, τµ (0 )] then dim α ≤α Eα ≤ τµ∗ (α).
µ 3. If α ≥ τµ (0+ ) then dim α ≥α Eα ≤ τµ∗ (α).
This is deduced from Theorem 1 of [14], Proposition 2.5 of [45], Theorem 1 of [39], αµ ⊂ Eαµ ∩ {x : αµ (x) = α}. Lemma 4.2 of [6] and the fact that E Definition 4. A positive Borel measure µ on [0, 1] is said to obey the multifractal forµ malism at α ≥ 0 if dµ (α) = dim(Eα ) = τµ∗ (α). 2.3. Conditions C1, C2(h) and C3(h). Definition 5. Let µ be a positive Borel measure with supp(µ) = [0, 1]. - Condition C1: There exists a constant B such that ∀j , ∀k = 0, .., bj − 1, µ(Ij,k ) ≥ b−Bj . - Condition C2(h): see Definition 6 in the next Sect. 3. µ ) - Condition C3(h): There exists a positive Borel measure mh on [0, 1] such that mh (E h ∗ > 0 and for every Borel set E ⊂ [0, 1] such that dim E < τµ (h), one has mh (E) = 0.
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3. Conditioned Ubiquity 3.1. Main result. Let us detail the assumptions that make Theorem 3 below work. The measure ν is built on the b-adic numbers, but the analysis of the initial measure µ may be naturally done using another base c. This is the case for instance for multinomial measures built in basis c, or for the c-adic Mandelbrot random multiplicative cascades. b
We shall thus deal with two bases simultaneously. When working in a basis b , Ij,k
b is the denotes the closed b -adic interval [kb −j , (k + 1)b −j ], and if x ∈ [0, 1), kj,x
integer k such that x ∈ [kb −j , (k + 1)b −j ) and Ijb (x) = I b
b j,kj,x
.
Assume that an atomless measure µ such that supp(µ) = [0, 1] is given, as well as two exponents α > 0 and β > 0 and an integer b ≥ 2. Our assumptions are as follows. H(α,β): (1) There exist two continuous non-decreasing functions ϕ and ψ defined on R+ such that: - ϕ(0) = ψ(0) = 0, r → r −ϕ(r) and r → r −ψ(r) are non-increasing near 0+ , and limr→0+ r −ϕ(r) = +∞. - ∀ε > 0, r → r ε−ϕ(r) is non-decreasing near 0 (which implies that r → r β/ξ −γ ϕ(r) is non-decreasing near 0 for β, γ , ξ > 0). - The next properties (2), (3) and (4) hold. (2) There exist an integer c ≥ 2, a constant M (depending on b and c) and a positive Borel measure m such that supp(m) = [0, 1] and −j m-a.e, ∃ n, ∀ j ≥ n, m Ijc (x) ≤ |Ijc (x)|β−ϕ(c ) ,
(7)
c c m-a.e, ∃ n, ∀ j ≥ n, PM (Ij,k ) holds for |k − kj,x | ≤ 2b2 c,
(8)
where PM (I ) is said to hold for an interval I when M −1 |I |α+ψ(|I |) ≤ µ I ≤ M|I |α−ψ(|I |) .
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Notice that β ≤ 1 since we work in R. (3) (Self-similarity property of m) For every closed c-adic subinterval I of [0, 1], let fI be the affine increasing mapping from I onto [0, 1]. There exists a measure mI on I , equivalent to the restriction of m to I , such that the measure mI ◦ fI−1 satisfies (7), and with the same exponent β. For every n ≥ 1, for every closed c-adic interval I of [0, 1], let c β−ϕ c−j |I | |Ij (x)| . EnI = x ∈ I : ∀ j ≥ n + logc (|I |−1 ), mI Ijc (x) ≤ |I |
The sets EnI form a non-decreasing sequence and by (7) n≥1 EnI is of full mI -measure. Let us define nI = inf n ≥ 1 : mI (EnI ) ≥ mI /2 . For x ∈ [0, 1) and j ≥ 0, let Ij (b, x) be the set of b-adic intervals of maximal length included in [kj,x c−j , (kj,x + 1/2)c−j ]. Then if L = [kb−j , (k + 1)b−j ] ∈
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Ij (b, x), for ξ > 1 let Lξ be the set of c-adic intervals of maximal length included in [kb−j , kb−j + b−j ξ ]. Finally we define
ξ
Ij (x) =
Lξ .
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L∈Ij (b,x)
(4) (Control of the speed of renewal nI and of the mass mI ) There exists a dense subset D of (1, ∞) such that for every ξ ∈ D, the property P(ξ ) holds, where P(ξ ) is: ξ for m-almost every x ∈ (0, 1), for every j large enough, there exists I ∈ Ij (x) such that nI ≤ logc |I |−1 )ϕ(|I |) and |I |ϕ(|I |) ≤ mI .
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Definition 6. Let µ be a positive Borel measure such that supp(µ) = [0, 1], and an integer b ≥ 2. C2(h) is said to hold for µ if H(h, τµ∗ (h)) holds. The assumptions on ϕ and ψ in (1) are purely technical, but non restrictive in practice. Assumption (2) allows to control m-almost everywhere the local behaviors of the analyzed measure µ and of the analyzing measure m. Assumption (3) emphasizes a selfsimilarity property of the analyzing measure m. Eventually, assumption (4) is a control (for some c-adic intervals I ) of mI and of the speed of renewal of the control (7) for the measure mI ◦ fI−1 . Assumptions (3) and (4) supply the monofractality property of the measures m used in [19, 32]. For these monofractal measures, there existβ > 0, C > 0 and r0 > 0 such that ∀ x ∈ supp(m), ∀ 0 < r ≤ r0 , C −1 r β ≤ µ B(x, r) ≤ Cr β . Theorem 3. Let µ be a positive Borel measure such that supp(µ) = [0, 1]. For ξ, M ≥ 1, α > 0 and ε = {εj }j ≥1 a non-negative sequence let Sξ,ε,M (α) =
[kb−j , kb−j + b−j (ξ −εj ) ].
n≥1 j ≥n k∈{0,... ,bj −1}:P (I b ) holds M j,k
Let α, β > 0 and suppose that H(α,β) holds. There exists M ≥ 1 such that for every ξ > 1, one can find a non-increasing sequence ε converging to 0 and a positive Borel measure mξ on [0, 1] such that mξ (Sξ,ε,M (α)) > 0, and for every x ∈ Sξ,ε,M (α), one has mξ B(x, r) lim sup β/ξ −5ϕ(r) < ∞. (12) r r→0+ Moreover, if ξ ∈ D then ε can be taken equal to {0}n≥1 . Corollary 1. If H(α,β) holds, then there exists M ≥ 1 such that for every ξ > 1, one can find a sequence ε such that Hf (Sξ,ε,M (α)) > 0, where Hf is the generalized Hausdorff f dimension H associated with the dimension (or gauge) function f : r → r β/ξ −5ϕ(r) . The mass distribution principle [20] implies that dim Sξ,ε,M (α) ≥ β/ξ , and for every Borel set E such that dim E < β/ξ , mξ (E) = 0.
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Theorem 2 is thus a consequence of the above corollary (the condition Qhψ is equivalent to the condition PM up to a small correction of the function ψ). The following property is used repeatedly in the sequel. Due to the assumption on ϕ and ψ, there exists a constant C > 0 such that for every 0 < r ≤ s ≤ 1, s −ϕ(s) ≤ Cr −ϕ(r) and s −ψ(s) ≤ Cr −ψ(r) .
(13)
Moreover, all along the proof, each time it occurs, C denotes a positive constant which depends only on α, β, ξ , ϕ and ψ. Before starting the proof, let us establish the following lemma. c ) holds for k ∈ {k c − Lemma 2. Let N ∈ N, and x ∈ (0, 1) such that PM (Ij,k j,x c + 2b2 c} for every j ≥ N . Then there exists a constant M (depending 2b2 c, . . . , kj,x only on b, c and µ) such that for every j ≥ N and every b-adic interval I of maximal c c−j , (k c + 1/2)c−j ], P (I ) holds. length contained in [kj,x M j,x
Proof. Let us fix j ≥ N and I a b-adic interval of maximal length contained in c c−j , (k c + 1/2)c−j ]. One has |I | ≥ c−j . Consequently, since I ⊂ I c (x) and [kj,x j,x j 2b2 since both PM (Ijc (x)) and (13) hold, there exists M ≥ 1 depending only on b, c and µ such that µ(I ) ≤ M |I |α−ψ(|I |) . Conversely, I contains at least one c-adic interval J of generation j = j +
[logc (2b2 )] + 1 which is distant from Ijc (x) by at most 2b2 c · c−j . By our assumption this implies that PM (J ) holds so µ(I ) ≥ M|J |α+ψ(|J |) ≥ M|J |α+ψ(|I |) if |I | is small |I |
enough (ψ is non-increasing near 0). Since |J | is bounded, there exists M ≥ 1 which
−1 α+ψ(|I |) depends only on b, c and µ such that µ(I ) ≥ M |I | . Proof. Let ξ > 1 and let {ξn }n≥0 ∈ DN be a non-decreasing sequence converging to ξ . To each ξn can be applied P(ξn ). Let M ≥ 1 be the constant computed in last Lemma 2. We shall construct step by step the sequence ε, a generalized Cantor set Kξ in Sξ,ε,M (α), and simultaneously the measure mξ on Kξ . c is also denoted I c . In the sequel, the closure of an interval Ij,k j,k - First step: The first generation of intervals involved in the construction of Kξ is taken as follows. Let us focus on ξ1 . L0 of EnLL0 of Let L0 = [0, 1]. By assumptions (2) and P(ξ1 ), there exist a subset E 0 L0 , m-measure larger than m/4 and an integer n L0 ≥ nL0 such that for every x ∈ E for every j ≥ n L0 , there exists I ∈ Ij 1 (x) such that (11) holds and simultaneously ξ
c c c ∀ j ≥ n L0 , PM (Ij,k ) holds for k ∈ {kj,x − 2b2 c, . . . , kj,x + 2b2 c}.
L0 possesses a Cantor-like structure: The set E L0 = E j ≥n L 0
c Ij,k .
(14)
(15)
L0 , I c =I c (x) k:∃x∈E j,k j
c c 1 (j ) = Ij,k L0 , Ij,k For j ≥ n L0 , let us define G : ∃x ∈E = Ijc (x) . c be a c-adic interval in G 1 (j ), x ∈ E L0 ∩ I c and a c-adic interval I ∈ I ξ1 (x) Let Ij,k j,k j b b . By such that (11) holds. Let IJ,K ∈ Ij (b, x) such that I ⊂ [Kb−J , Kb−J ξ1 ] ⊂ IJ,K b ) holds. Lemma 2, PM (IJ,K
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One remarks that one ensured by construction the existence of a constant C (dependξ c |ξ1 ≤ C|I |. ing on b and c only) such that ∀I ∈ Ij 1 (x), C −1 |I | ≤ |Ij,k c 1 (j ) is associated another (closed) smaller Thus, with every c-adic interval Ij,k in G c-adic interval I = Ijc ,k . Eventually (this is the key property to ensure that the generalized Cantor set will be included in Sξ,ε,M (α)) one remarks that Ijc ,k ⊂ [Kb−J , Kb−J + c for some Kb−J such that P (I b ) holds. b−J ξ1 ] ⊂ Ij,k M J,K c . Conversely, if a c-adic interval I can be written J for some We denote Ijc ,k = Ij,k c , for some choice larger c-adic interval J , one writes J = I . These small intervals Ij,k of j , will be construct Kξ . Let us define cthe firstc generation of c-adic intervals used to 1 (j ) . Notice that if I and I are two distinct elements of G1 (j ) = Ij,k : Ij,k ∈ G
G1 (j ), the distance between I and I is at least |I |/2. On the algebra generated by the elements of G1 (j ), a probability measure mξ is defined by mξ (I ) =
c Ij,k
m(I ) c . 1 (j ) m(Ij,k ) ∈G
By the assumption made on the measure m and (13), one has m(I ) ≤ |I |β−ϕ(|I |) ≤ C|I |β/ξ1 |I |−ϕ(|I |) ≤ C|I |β/ξ1 |I |−ϕ(|I |) . c L0 )≥m/4. Moreover, using the Cantor-like structure (15), I c ∈G 1 (j ) m(Ij,k ) ≥ m(E j,k As a consequence, ∀ I ∈ G1 (j ), mξ (I ) ≤ 4m−1 C|I |−ϕ(|I |) |I |β/ξ1 . By (1), j1 can be chosen large enough so that ∀ I ∈ G1 (j1 ), 4m−1 C ≤ |I |−ϕ(|I |) . We choose the c-adic elements of the first generation of the construction of Kξ as being those of G1 := G1 (j1 ). By construction, ∀ I ∈ G1 , mξ (I ) ≤ |I |β/ξ1 −2ϕ(|I |) .
(16)
- Second step: We construct the second generation of intervals. Consider ξ2 . For L of E LL such every L ∈ G1 , using assumptions (3) and (4), one can find a subset E n
L ≥ mL /4 and an integer n ≥ nL such that ∀ x ∈ E L , for every that mL E L ξ j ≥ n L + logc |L|−1 , there exists I ∈ Ij 2 (x) such that (11) holds and (as in (14)) c c ) holds for |k − kj,x | ≤ 2b2 c. ∀ j ≥ n L + logc |L|−1 , PM (Ij,k
(17)
c
L ,I c =I c (x) Ij,k , and one can define for every k:∃x∈E j ≥n L +logc |L|−1 j,k j L , I c = I c (x) . L (j ) = I c : ∃ x ∈ E j ≥ n L + logc |L|−1 the set G 2 j,k j,k j L Then, another set GL 2 (j ) of closed c-adic intervals is obtained from G2 (j ) by the 1 (j ) in the first step. Thus, with every same procedure as G1 (j ) is constructed from G c in G L (j ) is now associated a b-adic interval [Kb−J , (K + 1)b−J ] c-adic interval Ij,k 2 and another closed c-adic interval Ijc ,k with the following properties:
L = One has E
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c |ξ2 ≤ C|I c |, - their lengths satisfy C −1 |Ijc ,k | ≤ |Ij,k j ,k
c c b ) holds. −J −J −J ξ 2 ] ⊂ Ij,k and PM (IJ,K - Ij ,k ⊂ [Kb , Kb + b c and I c = I c . Here again, one writes Ijc ,k = Ij,k j,k j ,k
c c L L Let us define G2 (j ) = Ij,k : Ij,k ∈ G2 (j ) . On the algebra generated by the
elements I of GL 2 (j ), an extension of the restriction to the interval I of the measure mξ is defined by mξ (I ) =
c Ij,k
mL (I ) mξ (L). L c L (j ) m (Ij,k ) ∈G 2
By the assumption made on the measure mI , one shows that −ϕ |I | |I | β−ϕ |I | |L| |L| L β/ξ2 −β |I | ≤ C|I | ≤ C|I |β/ξ2 |L|−β |I |−ϕ(|I |) , |L| m (I ) ≤ |L| |L| L ) ≥ mL /4. mL (I c ) ≥ mL (E where (13) has been used. Moreover c L Ij,k ∈G2 (j )
j,k
Consequently, using (16) to get an upper bound for mξ (L), one obtains mξ (I ) ≤ mξ (L)
β/ξ1 −β−2ϕ(|L|) 4 β/ξ2 −β −ϕ(|I |) 4C|L| |L| |I | ≤ C|I | |I |β/ξ2 −ϕ(|I |) . mL mL
One can choose j2 (L) large enough so that for every integer j ≥ j2 (L), for every c-adic L −1 C|L|β/ξ1 −β−2ϕ(|L|) ≤ |I |−ϕ(|I |) . interval I in GL 2 (j ), 4m
Then, taking j2 = max j2 (L) : L ∈ G1 , and defining G2 = L∈G1 GL (j ), this
2 2 yields an extension of
mξ to the algebra generated by the elements of G1 G2 and such that for every I ∈ G1 G2 , since ξ2 ≥ ξ1 , mξ (I ) ≤ |I |β/ξ2 −2ϕ(|I |) .
(18)
- Third step: We end the induction. Assume that the first nth generations of intervals G1 , . . . , Gn are found for some
integer n ≥ 2. Assume also that a probability measure mξ on the algebra generated by 1≤p≤n Gp is defined and that the following properties hold (the fact that this holds for n = 2 comes from the two previous steps): (i) the elements of Gp are closed c-adic intervals and pairwise disjoint. With each I ∈ Gp is associated an interval I such that the I ’s, I ∈ Gp , are pairwise distinct c-adic intervals of the same generation, with C −1 |I |ξp ≤ |I | ≤ C|I |ξp for some universal constant C. If I and I are two distinct elements of Gp , the distance between I and I
is at least |I |/2. (ii) For every 2 ≤ p ≤ n, each I of Gp is a subinterval of an element L of element Gp−1 . Moreover, I ⊂ L, logc |I |−1 ≥ nL + logc |L|−1 and I ∩ EnLL = ∅. b = [kb−j , (k + (iii) For every 1 ≤ p ≤ n and I ∈ Gp , there is a b-adic interval Ij,k b −j −j −j −j ξ 1)b ) such that I ⊂ [kb , kb + b p ] ⊂ I and PM (Ij,k ) holds.
(iv) For every I ∈ 1≤p≤n Gp , mξ (I ) ≤ |I |β/ξp −2ϕ(|I |) ≤ |I |β/ξ −2ϕ(|I |) . (v) For every 1 ≤ p ≤ n − 1, L ∈ Gp , and I ∈ Gp+1 such that I ⊂ L, mξ (I ) ≤ 4mL −1 mξ (L)mL (I ).
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The construction of a generation Gn+1 of c-adic intervals and an extension of mξ to the algebra generated by the elements of 1≤p≤n+1 Gp such that properties (i) to (v) hold for n + 1 instead of n is done in the same way as when n = 1. For every n ≥ 1, let Jn = sup{J : ∃I ∈ Gn , ∃K, I ⊂ [Kb−J , Kb−J + b−J ξn ] ⊂ b ) holds} and J = 1. Then for every n ≥ 1, for every j ∈ [J I and PM (IJ,K 0 n−1 + 1, Jn ], one sets εj = ξ − ξn . By induction, and due to the separation property (i), we obtain a sequence (Gn )n≥1
and a probability measure mξ on σ I : I ∈ n≥1 Gn such that properties (i) to (v) hold for every n ≥ 2. Let us define Kξ = n≥1 I ∈Gn I. By construction, mξ (Kξ ) = 1 and because of property (iii) Kξ ⊂ Sξ,ε,M (α). Eventually, the measure mξ is extended to B([0, 1]) in the usual way: mξ (B) := mξ (B ∩ Kξ ) for every B ∈ B([0, 1]). - Last step: Proof of (12). If I ∈ Gn , we set g(I ) = n (the generation of the interval I ). Let us fix I an open subinterval of [0, 1] of length smaller than the lengths of the elements
of G1 , and assume that I ∩ Kξ = ∅. Let L be the element of largest diameter in n≥1 Gn such that I intersects at least two elements of Gg(L)+1 included in L. This implies that I does not intersect any other element of Gg(L) , and as a consequence mξ (I ) ≤ mξ (L). We distinguish three cases: • If |I | ≥ |L|, one has mξ (I ) ≤ mξ (L) ≤ |L|β/ξ −2ϕ(|L|) ≤ C|I |β/ξ −2ϕ(|I |) .
(19)
• If |I | ≤ c−nL −1 |L|, let L1 , . . . , Ld be the elements of Gg(L)+1 which intersect I . They are all sons of L. Property (v) above yields mξ (I ) =
d
4 L m (Li ). mL d
mξ (I ∩ Li ) ≤ mξ (L)
i=1
i=1
Let n be the unique integer such that c−n ≤ |I | < c−n+1 . Recall EnLL =
c Ij,k .
(20)
c ∩E L =∅ j ≥nL +logc (|L|−1 ) k:Ij,k n L
Due to property (i), d ≥ 2 implies |I | ≥ |Li |/2. Hence the scale of the intervals Li (which equals − logc |Li |) is larger than n − 1. Combining this with (ii) and (20), one
c can write that di=1 Li ⊂ k:I ∩I c ∩EnL =∅ In−1,k . There are at most 2 terms in the n−1,k L −n −1 L |L|, one has n − 1 ≥ nL + logc |L|−1 . Thus for previous union. Since |I | ≤ c c |I c | β−ϕ |In−1,k | |L| n−1,k c c L L ≤ each k such that I ∩ In−1,k ∩ EnL = ∅ one has m (In−1,k ) ≤ |L| β −ϕ |I | |L| |I | |I | C |L| , where C depends only on β. This yields |L| 4 L 4 mξ (I ) ≤ mξ (L) L m (Li ) ≤ mξ (L) L 2C m m i=1 C |I | β −ϕ(I ) ≤ mξ (L) L |I | . m |L| d
|I | |L|
β
|I | −ϕ |L|
|I | |L|
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We then use consecutively two facts. First by (iv), mξ (L) ≤ |L|β/ξ |L|−2ϕ(|L|) ≤ β(1−1/ξ ) C β/ξ −3ϕ(|I |) |I | C|L|β/ξ |I |−2ϕ(|I |) . This implies that mξ (I ) ≤ |I | |I | , mL |L| L −1 β/ξ −3ϕ(|I |) β(1−1/ξ ) which is smaller than Cm |I | |I | since r → r is bounded near 0. Then (4) yields an upper bound for mL −1 and mξ (I ) ≤ C|L|−ϕ(|L|) |I |β/ξ |I |−3ϕ(|I |) ≤ C|I |β/ξ |I |−4ϕ(|I |) .
(21)
• c−nL −1 |L| < |I | −n c L −1 |L| to cover I .
≤ |L|: one needs at most cnL +2 contiguous intervals of length For these intervals, the estimate (21) can be used. Thus for |I | small enough, and using again assumption (4), β/ξ −n −1 −4ϕ(c−nL −1 |L|) mξ (I ) ≤ CcnL +2 c−nL −1 |L| c L |L| ≤ CcnL |I |β/ξ |I |−4ϕ(|I |) ≤ C|L|−ϕ(|L|) |I |β/ξ |I |−4ϕ(|I |) ≤ C|I |β/ξ |I |−5ϕ(|I |) . The constant C > 0 does not depend on the interval I . Remembering (19) and (21), and using assumption (1), one gets that for every nontrivial subinterval L of [0, 1], mξ (L) ≤ C|L|β/ξ |L|−5ϕ(|L|) . 3.2. Examples of measures µ that satisfy C1, C2 and C3. We are going to describe four classes of statistically self-similar measures. For all these measures, property C1 follows easily from their study in the papers mentioned below. • Deterministic Gibbs measures. Let µ be a Gibbs measure associated with an H¨older potential φ in the dynamical system ([0, 1), T ), where T (x) = cx mod 1 with c an integer ≥ 2 (see [46]). The multifractal analysis of µ is performed for instance in [14, 48, 24]. In this case the function τµ is analytic, and the fact that C3(h) holds for all h of the form τµ (q), q ∈ R, is an easy consequence of the works mentioned above. The fact that C2(τµ (q)) holds for all q ∈ R is also simple in this case. Let q ∈ R. To see that H(τµ (q), τµ∗ (τµ (q))) holds, choose the analyzing measure m to be the Gibbs measure associated with the potential qφ (instead of φ for µ). The law of the iterated logarithm applied to the Birkhoff sums associated with φ with respect to m (see Chapter | log t| 1/2 for some 7 of [49]) show that property (2) holds with ϕ(t) = ψ(t) = C log|log log(t)|
C > 0. Also, if mI ◦ fI−1 = m, it is obvious that (3) and (4) hold, and the speed of renewal nI does not depend on I . • Random Gibbs measures. We consider the following particular class. We fix a potential φ as above, and a sequence ω = (ωn )n≥0 of independent random phases uniformly distributed in [0, 1]. If j ≥ 1 one denotes by ω(j ) the sequence (ωn )n≥j . For n ≥ 1 and k x ∈ [0, 1], let Sn (φ, ω)(x) = n−1 k=0 φ(T x + ωk ). It follows from the thermodynamic formalism for random transformations (see [35]) that, with probability one, the sequence of measures exp Sj (φ, ω)(x) φ,ω µj (dx) = dx [0,1] exp Sj (φ, ω)(u) du
converges weakly to a Gibbs measure µ. The fact that C3(h) holds for every h of the form τµ (q), almost surely, is a consequence of [36]. The stronger property “C3(h) holds almost surely for all h of the form τµ (q)” is established in [9]. The fact that, with probability one, H(τµ (q), τµ∗ (τµ (q))) holds for all q ∈ R is established in [11]. Given ω in
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the probability space such that µ(ω) is defined, for q ∈ R one takes m as a weak limit qφ,ω of a subsequence of the sequence (µj )j . In the same way, for j ≥ 1, one defines qφ,ω(j )
m(j ) as a weak limit of a subsequence of (µk )k . Then, if I is a c-adic interval of generation j , the measure mI is defined so that mI ◦ fI−1 = m(j ) . One gets (2), (3) 1 +η 1 and (4) with ψ(t) = ϕ(t) = | logb (t)|− 8 log | logb (t)| 2 for some η > 0. Moreover, since all the measures mI only depend on the generation of I , and not on I , [11] shows that the control (11) holds for all I of sufficiently large generation. • Canonical cascades measures. These measures are studied in particular in [40, 34, 29, 43, 5, 6, 11]. Let W be a positive random variable with expectation equal to 1, and let (WJ )J ∈I be a sequence of independent copies of W indexed by the set I of c-adic subintervals of [0, 1). The canonical cascade measure µ is the almost sure weak limit of the measure-valued martingale µj defined on [0, 1] by µj (dx) =
WJ dx.
c−j ≤|J |≤c−1 , x∈J
Let τ : q ∈ R → q − 1 − logc E(W q ). The condition τ (1− ) > 0 is necessary and sufficient to ensure that, with probability one, µ is non-degenerate, that is non-equal to zero [34]. We assume τ (1− ) > 0 and then define J , the interior of the interval {q ∈ R : τ (q)q − τ (q) > 0}. We assume that J is a neighborhood of [0, 1]. It is proved in the works mentioned above that, with probability one, τ and τµ coincide on the closure of J , and also that C3(h) holds for all h of the form τµ (q), q ∈ J . The following fact is established in [11]: For every q ∈ J , with probability one, H(τµ (q), τµ∗ (τµ (q))) holds. Also, with probability one, H(τµ (q), τµ∗ (τµ (q))) holds for almost-every q ∈ J (with respect to the Lebesgue measure). For q ∈ J , the analyzing measure m is obtained as µ but with the weights Wq,J = q WJ /E(W q ) instead of the WJ ’s, and the measure mI ◦fI−1 is the measure obtained as m, 1 +η − 21 I := W log | log(t)| 2 but with the weights Wq,J q,fI−1 (J ) . Moreover, ψ(t)=| log(t)| −κ and ϕ(t) = log | log(t)| for some η, κ > 0. Contrary to the case of random Gibbs measures, the measures mI are pairwise distinct. This reflects a higher degree of randomness in the construction: While only j i.i.d φ,ω random phases are needed to construct µj , bj independent copies of W enter in the definition of µj . This makes impossible to get uniformly over the c-adic intervals of sufficiently large generation the control (11) with a suitable function ϕ. • Compound Poisson cascades. Theses measures were recently introduced in [7]. Their construction is as follows (we do not enter the details here). Let θ > 0 and let be the measure on the strip R × (0, 1] given by its density (dtdλ) = θλ−2 dtdλ. Let S be a Poisson point process with intensity . With each M = (tM , λM ) ∈ S can be associated a positive integrable random variable WM in such a way that the WM ’s are i.i.d, and also independent of S. Then for (t, ε) ∈ [0, 1] × (0, 1] define Cε (t) = {(s, λ) ∈ R × [0, 1] : ε ≤ λ < 1, t − λ/2 < s ≤ t + λ/2}. The compound Poisson cascade measure µ on [0, 1] is the almost sure weak limit, as ε → 0, of the measure-valued martingale µε (dt) = εθ(E(W )−1)
M∈S∩Cε (t)
WM dt.
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Let τ (q) = −1 + q 1 + θ (E(W ) − 1) − θ (E(W q ) − 1). It is shown in [11] that under the same assumptions on τ as for canonical cascades measures, one has for µ formally the same conclusions on C2(h) and C3(h) as for these measures. Extensions of µ are proposed in [4] and in [8], where for instance the following extended construction is developed: If φ is chosen as for the Gibbs measures, let µε (dt) = ε
θ E(W ) [0,1] exp(φ(t) dt−1
WM exp φ λ−1 M (t − tM − λM /2) dt.
M∈S∩Cε (t)
The techniques developed for µ in [11] also hold for the limit of µε . 4. Proof of Theorem 1 The claims on the multifractal formalism and the identification of qc (ν) and hc (ν) are postponed to Sect. 5. Before stating some results, let us remark that one has for some constant C independent of j , k, and µ, k , ν(Ij,k ) ≥ ν({kb−j }) ≥ j −2 µ(Ij,k ), bj µ(Ij ,k ) µ(Ij,k ) µ(Ij,k ) k . = ≤C if j irreducible, ν(Ij,k ) =
2
2 b j j j
for every
j ≥j k b−j ∈Ij,k
(22) (23)
j ≥j
4.1. First properties of ν. Remember the definition (5) of ξx . Obviously, if x is a badic number kb−j and if Kb−J is its irreducible representation, either hν (x) = 0 if µ(IJ,K ) > 0, or hν (x) = +∞ if µ(IJ,K ) = 0. Lemma 3. Assume C1 for µ. If x ∈ supp(µ) and ξx = +∞, hν (x) = 0. Proof. Let be B be as in C1. Let M > B. Since ξx = +∞, there exists an infinite number of b-adic numbers kb−j with j ≥ J such that |kb−j − x| ≤ b−j M . Let k0 b−j0 be such a b-adic number. Let J0 = [Mj0 ] − 2 and let K0 be such that K0 b−J0 = k0 b−j0 . Since |k0 b−j0 −x| ≤ b−j0 M , one has |K0 b−J0 −x| ≤ b−(J0 +1) , and thus K0 b−J0 ∈ B(x, b−J0 ). Using (22), for some constant C depending on B and M one has ν(B(x, b−J0 )) ≥ ν({k0 b−j0 }) ≥ j0−2 µ(Ij0 ,k0 ) ≥ j0−2 b−Bj0 ≥ CJ0−2 b− M J0 . B
There exists an infinite number of integers J0 such that last inequality holds, thus hν (x) ≤ B/M. This remains true for any M > B, thus hν (x) = 0. µ
Proposition 2. Let x ∈ Eα for some α ≥ 0, and assume that its approximation rate by the b-adic numbers ξx is finite. Then ξαx ≤ hν (x) ≤ α. Proof. Let ε > 0. Let us first obtain an upper bound for hν (x). By definition of α, there exists an infinite number of integers j0 such that max µ(Ij−0 (x)), µ(Ij0 (x)), µ(Ij+0 (x)) ≥ b−j0 (α+ε) . Let j0 be such an integer, and let us
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then find a lower bound for ν(B(x, b−(j0 −1) )). It is obvious that Ij−0 (x)∪Ij0 (x)∪Ij+0 (x) ⊂ B(x, b−(j0 −1) ). Thus using (22), one gets ν(B(x, b−(j0 −1) )) ≥ max(ν(Ij−0 (x)), ν(Ij0 (x)), ν(Ij+0 (x))) ≥ j0−2 max(µ(Ij−0 (x)), µ(Ij0 (x)), µ(Ij+0 (x))) ≥ j0−2 b−j0 (α+ε) . −(j −1)
)) ν(B(x,b This implies hν (x) = lim inf j →+∞ log ≤ α + ε. This remains true for log |B(x,b−(j −1) )| every ε > 0, hence the result. Let us move to the lower bound. By definition of ξx , there exists J such that j ≥ J
µ implies ∀k, |kb−j − x| ≥ b−j (ξx +ε) . Moreover, x ∈ Eα , thus there exists a scale J
− +
such that j ≥ J implies max(µ(Ij (x)), µ(Ij (x)), µ(Ij (x))) ≤ b−j (α−ε) . One sets J = max([2(ξx + 1)J ], [2(ξx + 1)J
]). Let j0 ≥ J , and consider B(x, b−j0 ). For every j ≥ j0 + 1, one has µ(Ij,k ) ≤ µ(Ij−0 (x)) + µ(Ij0 (x)) + µ(Ij+0 (x)) ≤ 3 b−j0 (α−ε) . kb−j ∈B(x,b−j0 )
since B(x, b−j0 ) ⊂ Ij−0 (x) ∪ Ij0 (x) ∪ Ij+0 (x). This yields ν(B(x, b−j0 )) = ν ≤ν Thus for any x ∈
µ Eα
k
j0 ,x bj0
k
j0 ,x bj0
+ν +ν
k k
j0 ,x + 1 bj0 j0 ,x + 1 bj0
+
j ≥j0 +1
+
j ≥j0 +1
1 j2
µ(Ij,k )
kb−j ∈B(x,b−j0 )
1 3 b−j0 (α−ε) . j2
and for j0 large enough, one has
ν(B(x, b−j0 )) ≤ ν({kj0 ,x b−j0 }) + ν({(kj0 ,x + 1)b−j0 }) + Cj0−1 b−j0 (α−ε) .
(24)
This inequality will later be of great importance. We distinguish three cases: - if kj0 ,x is a multiple of b: kj0 ,x b−j0 can be written as an irreducible fraction K0 b−J0 with J0 < j0 . Since |K0 b−J0 − x| ≤ b−j0 ≤ b−(J0 +1) , K0 b−J0 is the b-adic number that is the closest to x at scale J0 . The integer J has been chosen large enough so that the reduced scale J0 is greater than J . Hence one gets that |K0 b−J0 − x| ≥ b−J0 (ξx +ε) . Thus b−J0 (ξx +ε) ≤ |K0 b−J0 − x| ≤ b−j0 , which implies j0 ≤ J0 (ξx + ε). Moreover, since J0 ≥ J
, one obtains µ(IJ0 ,K0 ) ≤ b−J0 (α−ε) . One can now get an upper bound for ν({kj0 ,x b−j0 }). Indeed, for some constant Cξx that depends on ξx , α−ε j −2 µ(IJ0 ,K0 ) ≤ CJ0−1 b−J0 (α−ε) ≤ Cξx j0−1 b−j0 ξx +ε . ν({kj0 ,x b−j0 }) ≤ j ≥J0
- if kj0 ,x + 1 is a multiple of b: the same arguments apply also here, and ν({(kj0 ,x +
1)b−j0 }) ≤ Cξx j0−1 b−j0 ξx +ε . - if kj0 ,x (or kj0 ,x ) is not a multiple of b: then by (23) one has ν({kj0 ,x b−j0 }) ≤ Cj0−1 b−j0 (α−ε) (or ν({(kj0 ,x + 1)b−j0 }) ≤ Cj0−1 b−j0 (α−ε) . α−ε
Eventually, ν(B(x, b−j0 )) ≤ 2Cξx j0−1 b−j0 ξx +ε + Cj0−1 b−j0 (α−ε) ≤ Cj0−1 b−j0 ξx +ε . As a consequence, hν (x) ≥ ξα−ε , and this is true ∀ε > 0, hence the result. x +ε α−ε
α−ε
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4.2. Decomposition of Ehν . The following sets are needed. Definition 7. Let µ be a positive Borel measure, and α ≥ 0, ξ ≥ 1 be two real numbers. Let ε > 0. For every point x, the property L(α, ξ, ε) is said to hold at x if there exist η ≤ ε, and an infinite number of b-adic numbers kb−j that verify b−j (α+η) ≤ µ([kb−j , (k + 1)b−j )) ≤ b−j (α−η) and |kb−j − x| ≤ 2 b−j ξ . Let now h ≥ 0. The set Fh is defined by ∀ε > 0, ∃ α ≥ 0, ξ ≥ 1 such that Fh = x : α ≤ h + ε and L(α, ξ, ε) holds at x .
(25)
(26)
ξ
It is obvious to verify that for any 0 ≤ h ≤ h , Fh ⊂ Fh .
Proposition 3. Let h > 0. One has Ehν = Fh \ h 0, and (α, ξ ) such that αξ ≤ h + ε and L(α, ξ, ε) holds at x. For some η < ε, denote by kn b−jn an infinite sequence of b-adic numbers such that b−jn (α+η) ≤ µ([kn b−jn , (kn + 1)b−jn )) ≤ b−jn (α−η) and |kn b−jn − x| ≤ 2 b−jn ξ . Since ν(B(x, 2b−jn ξ )) ≥ − log jn2 log 2b−jn ξ
hν (x) ≤
jn (α+η) jn ξ −log 2 . The α+η ξ ≤ h + 2ε.
+
1 µ([kn b−jn , (kn jn2
+ 1)b−jn )), one gets
right term tends to
α+η ξ
log ν(B(x,2b−jn ξ )) log 2b−jn ξ
≤
when jn → +∞, hence ∀ε > 0,
The following proposition is important to prove Proposition 3 and also to find the upper bound in the next section. Proposition 4. Let h > 0 and x ∈ Ehν . Assume C1 holds for µ. Then x ∈ Fh . Proof. Let ε > 0, and x ∈ Ehν . We want to show that there exists a couple (α, ξ ) such that αξ ≤ h + ε and L(α, ξ, ε) holds at x. Let αx > 0 the unique exponent such that µ x ∈ Eαx (remember that by Proposition 2, αx = 0 ⇒ hν (x) = 0). 1. ξx = 1: by Proposition 2, one has h = αx . One can take ξ = 1, α = h + ε. Indeed, if x ∈ Eµh , there exists an infinite number of intervals I ∈ {Ij− (x), Ij (x), Ij+ (x)} such that b−j (h+ε) ≤ µ(I ) ≤ b−j (h−ε) . Such intervals I satisfy (25). 2. ξx > 1 and h = αx : the arguments of item 1 apply with ξ = 1 and α = αx + ε. 3. ξx > 1 and h < αx : we assume that ε is small enough so that h + ε < αx − ε. Remark that if b-adic numbers that satisfy (25) exist, then k = kj,x or k = kj,x + 1. By definition of ξx , there exists a scale J such that j ≥ J implies ∀k, |kb−j − x| ≥ µ −j (ξx + 3ε ) b , and since x ∈ Eαx , one can similarly impose J large enough so that for every ε j ≥ J , max(µ(Ij− (x)), µ(Ij (x)), µ(Ij+ (x))) ≤ b−j (αx − 3 ) . Since x ∈ Ehν , there exists an infinite number of integers jn ≥ J such that ν(B(x, ε b−jn )) ≥ b−jn (h+ 3 ) . Consider one of these jn . Since h + 3ε < αx − 3ε , (24) yields for jn large enough and for some constant C depending on x, h and αx , C −1 b−jn (h+ 3 ) ≤ ν({kjn ,x b−jn }) + ν({(kjn ,x + 1)b−jn }). ε
(27)
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Remark that one of kjn ,x and kjn ,x + 1 must be a multiple of b. Indeed, otherwise we would have by (23) ν({kjn ,x b−jn }) + ν({(kjn ,x + 1)b−jn }) ≤ j2n µ(Ijn (x)) ≤ 2 −jn (αx − 3ε ) . jn b
- If kjn ,x
Thus if ε is small enough so that αx − ε > h + ε, this is impossible. is a multiple of b: then (kjn ,x + 1)b−jn is irreducible, and by (23) ν({(kjn ,x + 1)b−jn }) ≤ Cjn−1 µ(Ij+n (x)) ≤ Cjn−1 b−jn (αx − 3 ) . ε
Thus (27) can be rewritten for jn large enough ν({kjn ,x b−jn }) ≥ C −1 b−jn (h+ 3 ) , up to a modification of the value of the constant C. Let us write kjn ,x b−jn = Kn b−Jn , where Kn is not a multiple of b. By construction |Kn b−Jn − x| = b−Jn ξn , where ξn ≥ 1. Moreover, by (23), ε
CJn−1 µ(IJn ,Kn ) ≥ ν({kjn ,x b−jn }) ≥ C −1 b−jn (h+ 3 ) = C −1 b−Jn ξn (h+ 3 ) . ε
ε
Thus for jn large enough, µ(IJn ,Kn ) = b−Jn αn where αn ≤ ξn (h + 2 3ε ). Eventually, for the b-adic number Kn b−Jn and its corresponding interval IJn ,Kn , (25) is satisfied with the couple (αn , ξn ). Remark that ξn ∈ [1, ξx + 3ε ] (because Jn ≥ J ) , and that αξnn ≤ h + 2 3ε < h + ε. - If kjn ,x + 1 is a multiple of b: the same arguments as above also apply here. Since C1 is satisfied, by Definition 5 there exists B such that for every j and k, µ(Ij,k ) ≥ b−Bj . One can thus extract an infinite subsequence of b-adic numbers Kn b−Jn that verify (25) with (αn , ξn ) ranging in the square S = [αx − 3ε , B] × [1, ξx + 3ε ] and satisfying αn ε ξn ≤ h + 2 3 . One can extract from (αn , ξn )n a subsequence (αφ(n) , ξφ(n) ) converging to some value (α0 , ξ0 ), that also satisfies αξ00 ≤ h + 2 3ε . Now choose η small enough such α0 +η ≤ h + ε, define ξ0 = max(1, ξ0 − η) and consider the square Sη = that max(1,ξ 0 −η) [α0 − η, α0 + η] × [ξ0 , ξ0 + η]. There exists a scale N such that n ≥ N implies (αφ(n) , ξφ(n) ) ∈ Sη . By construction, for every n ≥ N , one has b−Jφ(n) (α0 −η) ≥
µ(IJφ(n) ,Kφ(n) ) ≥ b−Jφ(n) (α0 +η) and |Kφ(n) b−Jφ(n) − x| = b−Jφ(n) ξφ(n) ≤ b−Jφ(n) ξ0 . Hence
L(α0 , ξ0 , ε) holds at x.
Proof (of Proposition 3). The last proposition shows that Ehν ⊂ Fh (one also has Ehν ⊂ ν
h >h Fh ).
Moreover, applying Lemma 4 to h < h yields Eh ∩ Fh = ∅. Hence ν Eh ⊂ Fh \ h
Conversely, let
x ∈ Fh \ h
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Proof. Let h > τµ (0+ ), and x ∈ Ehν . Let α be the unique exponent such that x ∈ Eα .
µ By Proposition 2, h = hν (x) ≤ α, hence x ∈ α ≥h Eα . Finally, by Proposition 1,
µ dim Ehν ≤ dim α ≥h Eα ≤ τµ∗ (h). µ
To prove the upper bound when h ≤ τµ (0+ ), one uses the next technical lemma. Lemma 5. Assume C1 holds for µ. Let f : R → R be a positive strictly increasing continuous function such that lim+∞ f (x) = +∞. Let us define ∀ε > 0, ∃ α ≥ 0, ξ ≥ 1 such that Gh (f ) = x : f (α) . ξ ≤ h + ε and L(α, ξ, ε) holds at x Then dim Gh (f ) ≤ h supα:f (α)≥h
supα ≤α τµ∗ (α ) . f (α) ∼
∼
ε , and α i be such that f (α i ) = Proof. Let ε > 0, and for every i ∈ N, let ξi = 1 + i 2h ∼
ξi (h + 2ε). Remark that ξi and α i have been chosen so that for ε > 0 small enough, for every ξ ∈ [ξi , ξi+1 ], one has ∼
f (α i ) = ξi (h + 2ε) ≥ ξ(h + ε).
(28)
∼
Thus let ε > 0 such that (28) holds, let αi = α i + ε, and let us define the sets Tαi ,ξi by Tαi ,ξi = [kb−j − 2 b−j ξi , kb−j + 2 b−j ξi ]. (29) J ≥0 j ≥J k:µ(Ij,k )≥b−j αi
Any point of Tαi ,ξi is infinitely many often close at rate ξi to a b-adic number kb−j that verifies µ(Ij,k ) ≥ b−j αi . By definition of Gh (f ), every x ∈ Gh (f ) belongs to Tαi ,ξi with i the unique integer such that ξ ∈ [ξi , ξi+1 ). One thus gets the inclusion Gh (f ) ⊂ i∈N Tαi ,ξi . It is time to use Lemma 1 to obtain an upper bound for the dimension of a set Tα,ξ . Indeed, let α > 0, ξ ≥ 1 and ε < ε. By Lemma 1 applied to η = ε /2 and ε = ε , one g gets that for j large enough (one also uses that dµ (α) is always smaller than τµ∗ (α), see Proposition 1) log # k : µ(Ij,k ) ≥ b−j α ≤ sup dµg (α ) + ε ≤ sup τµ∗ (α ) + ε . log bj α ≤α+ε /2 α ≤α+ε /2 We denote
sup
α ≤α+ε /2
τµ∗ (α ) + ε by τα,ε . Let us get the upper bound for the Hausdorff
dimension of Tα,ξ . Let d >
τα,ε
ξ . This set Tα,ξ
is covered by
j ≥J
k:µ(Ij,k )≥b−j α [kb
−j −
b−j ξ , kb−j + b−j ξ ], and |[kb−j − b−j ξ , kb−j + b−j ξ ]|d ≤ C bj τα,ε b−j dξ ≤ Cb−J (τα,ε −dξ ) , j ≥J k:µ(Ij,k )≥b−j α
j ≥J
where C is a constant that does not depend on d or J . This double sum goes to zero when J → +∞, and the d-dimensional Hausdorff measure of Tα,ξ is finite for every τ
τα,ε
d > α,ε ξ . Thus the Hausdorff dimension of Tα,ξ is less than ξ . This remains true for
Multifractal Additive and Multiplicative Chaos
493
any ε > 0, so, using the continuity of τµ∗ , one gets dim Tα,ξ ≤
The inclusion Gh (f ) ⊂ i∈N Tαi ,ξi implies dim Gh (f ) ≤ sup (dim Tαi ,ξi ) ≤ sup i∈N
≤ (h + 2ε) sup
=
supα ≤α τµ∗ (α ) . ξ
supα ≤αi τµ∗ (α )
i∈N supα ≤αi τµ∗ (α )
f (αi )
i∈N
inf ε τα,ε
ξ
ξi ≤ (h + 2ε)
sup
supα ≤α τµ∗ (α ) f (α)
α:f (α)≥h
,
where the range of α’s is f (α) ≥ h since f (αi ) is by definition always greater than h. Letting ε go to zero yields the conclusion. Proposition 6. Assume C1. If h ∈ (0, τµ (0+ )), dim(Ehν ) ≤ h supu≥h
τµ∗ (u) u .
Proof. Let h > 0, and x ∈ Ehν . Since x ∈ Ehν , by Proposition 4 x ∈ Fh (indeed, Fh corresponds in view of Lemma 5 to Gh (f ) where the function f is the identity sup
τ ∗ (α )
f (x) = x). Hence by Lemma 5, dim Ehν ≤ h supα≥h α ≤αα µ . Let us now simplify this formula. Remember (3) and simply write qc for qc (µ) and
τµ∗ (α) α is concave, and τµ∗ (hc ) τµ∗ (α) τµ∗ (hc ) maximum at hc , with qc = hc . Hence ∀α, α ≤ hc . Also, when α supα ≤α τµ∗ (α ) τ ∗ (τ (0+ )) τ ∗ (τ (0+ )) τ ∗ (hc ) ≤ µ µα ≤ µτ µ(0+ ) ≤ µhc = qc . α
hc for hc (µ). Since τµ∗ is concave, the function α →
reaches its ≥ τµ (0+ ),
µ
Two cases can thus be distinguished - hc < h < τµ (0+ ): If h ≤ α ≤ τµ (0+ ), supα ≤α τµ∗ (α ) = τµ∗ (α), so τµ∗ (α) α .
If α
≥
τµ (0+ ),
supα ≤α τµ∗ (α ) α
≤
τµ∗ (τµ (0+ )) . τµ (0+ )
supα ≤α τµ∗ (α ) α
Hence one gets
τ ∗ (α) τ ∗ (α) dim Ehν ≤ h supτµ (0+ )≥α≥h µα = h supα≥h µα . - 0 < h ≤ hc : If α ≥ hc , the same arguments as above still work. If h ≤ τ ∗ (α) τ ∗ (α) τ ∗ (hc ) supα ≤α τµ∗ (α ) ≤ µα ≤ µhc = qc . Thus dim Ehν ≤ hqc = h supα≥h µα . α
Let us now verify that the upper bound h supu≥h
τµ∗ (u) u
=
α < hc ,
coincides with the one an-
τ ∗ (u) nounced in Theorem 1. When h ≤ hc , supu≥h µu = qc , and the upper bound becomes τ ∗ (u) τ ∗ (h) τ ∗ (α) dim Ehν ≤ qc h. When h ≥ hc , supu≥h µu = µh (the mapping α → µα is τ ∗ (h) non-increasing when α ≥ hc ), hence dim Ehν ≤ h µh = τµ∗ (h).
A simple adaptation of the last proof yields the following corollary
Corollary 2. Let h ∈ [0, τµ (0+ )], and Fh be the set (26). Then dim Fh ≤ qc h. 4.4. Lower bound for the multifractal spectrum. For every j , k and ξ , one denotes (ξ ) Ij,k = [kb−j , kb−j + b−j ξ ]. Here again, qc (µ) and hc (µ) are simply denoted by qc and hc . Proposition 7. Let µ be a measure satisfying C2(hc ). Then ∀ξ ≥ 1, dim Ehνc /ξ ≥ τµ∗ (hc )/ξ .
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J. Barral, S. Seuret
Proof. Let ξ > 1, h = hc /ξ and d = τµ∗ (hc )/ξ . We apply Theorem 2. There exist a nonnegative sequence ε converging to 0, a non-negative continuous function ψ on R+ such that ψ(0) = 0, and a positive Borel measure mξ such that mξ (Sξ,ε,ψ (hc )) > 0 and for every Borel set E with dim E < d, mξ (E) = 0. Recall also that Ehν = Fh \ h
Using Corollary 2, for every i ≥ [h−1 ] + 1, dim Fh− 1 < qc h. This implies, by Theorem i
2, that mξ ( i≥[h−1 ]+1 Fh− 1 ) = 0. i Also one verifies that Sξ,ε,ψ (hc ) ⊂ Fh , since every point of Sξ,ε,ψ (hc ) satisfies L(hc , ξ −ε2 , ε) for every ε > 0 small enough. This implies that mξ (Ehν ) ≥ mξ (Sξ,ε,ψ (hc )) > 0, and thus that dim Ehν ≥ d. If ξ = 1, since C2(hc ) implies C3(hc ), see the proof of Proposition 8. Proposition 8. Let µ be a positive Borel measure supported by [0, 1], and let us assume that C3(h) holds for some h ≥ hc . Then dν (h) = dim Ehν ≥ τµ∗ (h). , the measure mh provided by C3(h) and ε > 0. Proof. Consider E h µ : ξx = ξ }. The same µ = {x ∈ E Let ξ > 1. Let us estimate the dimension of E h,ξ h lines of computations as in Lemma 5 show that, for every J , µ
ξ >ξ
µ ⊂ E h,ξ
j ≥J
k∈{0,... ,bj −1}: b−j (h+ε) ≤µ(Ij,k )≤b−j (h−ε)
[kb−j − b−j (ξ −ε) , kb−j + b−j (ξ −ε) ].
Applying Lemma 1 with η = ε gives # k : b−j (h+ε) ≤ µ(Ij,k ) ≤ b−j (h−ε) ≤ g
bj (supmax(β−ε,0)≤α ≤α+ε dµ (α )+ε) . One then uses that ∀α ∈ [max(β −ε, 0), α +ε], dµ (α ) ≤ τµ∗ (α ). Let us denote τh,ε = supmax(β−ε,0)≤α ≤α+ε τµ∗ (α ) + ε.
µ ≤ τh,ε . This is true ∀ε > 0, Using the covering, one deduces that dim ξ >ξ E ξ −ε h,ξ
µ ≤ τµ∗ (h)/ξ . hence using the continuity of τµ∗ on its support, dim ξ >ξ E h,ξ
µ \ i≥2 ξ >1+i −1 E µ . For i ≥ 2, dim ξ >1+i −1 E µ < τµ∗ (h), µ = E Let E h,1 h h,ξ h,ξ
µ ) = mh (E h ), which is > 0 by C3(h). µ ) = 0. Hence mh (E and thus mh ( ξ >1+i −1 E h,1 h,ξ µ all verify ξx = 1. Thus by Proposition 2, hν (x) = h. The points x belonging to E h,1 µ ⊂ E ν . This yields mh (E ν ) > 0 and dim E ν ≥ τµ∗ (h). Hence E h h h h,1 g
4.5. How to get the general case?. Reading attentively the arguments developed to study the measures ν = ν0,1 yields the proof of Theorem 1’. One can verify that ∀x ∈ [0, 1], γ +σ hµ (x) ≤ hνγ ,σ (x) ≤ γ + σ hµ (x), and then use the sets ξ Fh,γ ,σ = x :
∀ε > 0, ∃ α ≥ 0, ξ ≥ 1 such that ≤ h + ε and L(α, ξ, ε) holds at x
γ +σ α ξ
instead of the sets Fh and Lemma 5 applied with f (α) = γ + σ α to get upper and lower bounds for the multifractal spectrum of νγ ,σ . This is left to the reader.
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5. Additional Properties The reader can check that the upper multifractal spectrum defined by d ν (h) = dim{x : ν(B(x,r)) lim supr→0+ log log |B(x,r)| = h} is equal to the one of µ (when C3(h) holds). We set αmax = sup{α : τµ∗ (α) > 0}. Proposition 9. Let µ be a positive Borel measure on [0, 1], and let γ ≥ 0 and σ ≥ 1. h −γ Let qγ ,σ and hγ ,σ be defined as in Theorem 1’. Assume that C2( γ ,σσ ) holds, and that −γ
h
γ ,σ , αmax ). The measure νγ ,σ satisfies the C3( h−γ σ ) holds for every exponent h ∈ [ σ multifractal formalism at every h such that τν∗γ ,σ (h) > 0.
Proof. We give the proof in the case of ν, i.e. when γ = 0 and σ = 1. Here again, qc (µ) and hc (µ) are simply denoted by qc and hc . Let us compute τν . - if q < 0 : For every (j, k), by (22) ν(Ij,k ) ≥ j −2 µ(Ij,k ), which shows that bj −1 q bj −1 q −2q k=0 ν (Ij,k ) ≤ j k=0 µ (Ij,k ). Hence τν (q) ≥ τµ (q). Moreover, when C3(h) holds at h ∈ [τµ (0+ ), αmax ), τµ∗ (h) = dν (h) ≤ τν∗ (h). If this holds on a dense set of exponents h ∈ [τµ (0+ ), αmax ), by inverse Legendre transform one gets τν (q) ≤ τµ (q) for every q < 0 with τµ (q + ) ≤ αmax . The equality follows. J −1 q ν (IJ,K ) = Kb−J irreducible ν q (IJ,K ) + - if 0 < q < qc : Let J ≥ 2. One has bK=0 q −J is irreducible, one uses (23) to get ν q (I J,K ) ≤ K multiple of b ν (IJ,K ). When Kb 1 q q C J q µ (IJ,K ). When K is a multiple of b, let kb−j be its unique irreducible representation (0 ≤ j ≤ J − 1). As already noticed before, in this case ν(IJ,K ) = ν({kb−j }) +
j ≥J +1
≤ Cj −2 µ(Ij,k ) +
1 j 2
µ(Ij ,k )
k :Ij ,k ⊂IJ,K
Cj
−2
µ(IJ,K ) ≤ j −2 µ(Ij,k ) + CJ −1 µ(IJ,K ).
j ≥J +1
q Since q < 1, one gets ν q (IJ,K ) ≤ C q j12 µ(Ij,k ) + J1 µ(IJ,K ) ≤ C q j 12q µq (Ij,k ) + q 1 q 1 1 q ≤ K :IJ,K ⊂Ij,k µ(IJ,K ) J q µ (IJ,K ) . The term j 2q µ (Ij,k ) is bounded by j 2q 1 q K :I ⊂Ij,k µ (IJ,K ). This results yields jq J,K
ν q (IJ,K ) ≤
0≤K
Cq q q µ (I ) + C J,K Jq J 0≤K
K multiple of b, and kb−j its irreducible representation
µq (IJ,K ) . jq
K :IJ,K ⊂Ij,k
Each irreducible b-adic number kb−j with 0 ≤ j ≤ J − 1 appears one time in the above double sum. Conversely, for a given integer K ∈ {0, . . . , bJ − 1} and for each scale j , there exists only one irreducible b-adic number kb−j such that IJ,K ⊂ Ij,k . −1 1 bJ −1 q Hence, the double sum can be bounded by Jj =1 K=0 µ (IJ,K ), and eventually by jq bJ −1 q bJ −1 q bJ −1 q J K=0 µ (IJ,K ). So K=0 ν (IJ,K ) ≤ CJ K=0 µ (IJ,K ), where C is a constant independent of µ and J . This implies τν (q) ≥ τµ (q). On the other hand, when C3(h) holds on a dense set of values of h ∈ [hc , τµ (0+ )], at these exponents one has τν∗ (h) ≥ dν (h) = τµ∗ (h), which yields by inverse Legendre transform τν (q) ≤ τµ (q) for every q ∈ [0, qc ].
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- if q ≥ qc : Let us distinguish two cases. If qc = 1, then Theorem 1 yields dν (h) = h for h ∈ [0, τµ (1− )]. Hence τν∗ (h) ≥ h for h ∈ [0, τµ (1− )], but one always has τν∗ (h) ≤ h, hence τν∗ (h) = h, which gives by inverse Legendre transform τν (q) = 0 for q ≥ qc = 1, as well as qc (ν) = 1 and hc (ν) = τµ (1− ). If qc < 1, notice that τν (1) = 0 and τν (qc ) = τµ (qc ) = 0 since τν = τµ near qc− . Then the concavity and the monotonicity of τν force τν (q) = 0 for q ≥ qc . Moreover, since τν∗ (h) = qc h for h ∈ [0, hc ], one has qc (ν) = qc and hc (ν) = hc . Theorem 1 and the above identification of τν show that under our assumptions, dν (h) = τν∗ (h) for every h ∈ [0, αmax ). Finally, it can also be verified using [6] that under the assumptions of Proposition 9, the multifractal formalisms defined in [14, 13] and [45] are verified if one uses the level µ µ are considered. sets Eh . The formalisms do not hold if the sets E h References 1. Arbeiter, M., Patzschke, N.: Random self-similar multifractals. Math. Nachr. 181, 5–42 (1996) 2. Arnol’d, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Translated by J. Sz¨ucs, New York: Springer-Verlag, 1983 3. Aversa, V., Bandt, C.: The multifractal spectrum of discrete measures. Acta Uni. Caro.- Math. Phys. 31(21), 5–8 (1990) 4. Bacry, E., Muzy, J.-F.: Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236, 449–475 (2003) 5. Barral, J.: Continuity of the multifractal spectrum of a random statistically self-similar measures. J. Theor. Probab. 13, 1027–1060 (2000) 6. Barral, J., Ben Nasr, F., Peyri`ere, J.: Comparing Multifractal Formalisms: the neighboring boxes condition. Asian J. Math. 7, 149–166 (2003) 7. Barral, J., Mandelbrot, B.: Multifractal products of cylindrical pulses. Probab. Theory Relat. Fields 124(3), 409–430 (2002) 8. Barral, J., Mandelbrot, B.: Random multiplicative multifractal measures. In: Fractal Geometry and Applications, Proc. Symp. Pure Math., Providence, RI: AMS, 2004 9. Barral, J., Coppens, M.-O., Mandelbrot, B.: Multiperiodic multifractal martingale measures. J. Math. Pures Appl. (9) 82, 1555–1589 (2003) 10. Barral, J., Seuret, S.: From multifractal measures to multifractal wavelet series. Preprint (2002) 11. Barral, J., Seuret, S.: Inside singularity sets of random bibbs measures, and Renewal of singularity sets of statistically self-similar measures. Preprint (2004) 12. Barral, J., Seuret, S.: Function series with multifractal variations. Math. Nachr. 274–275, 3–18 (2004) 13. Ben Nasr, F.: Analyse multifractale de mesures. C. R. Acad. Sci. Paris S´erie I 319, 807–810 (1994) 14. Brown, G., Michon, G., Peyri`ere, J.: On the multifractal analysis of measures. J. Stat. Phys. 66(3–4), 775–790 (1992) 15. Cawley, R., Mauldin, R.D.: Multifractal decompositions of Moran fractals. Adv. Math. 92, 196–236 (1992) 16. Collet, P., Lebowitz, J.L., Porzio, A.: The dimension spectrum of some dynamical systems. J. Stat. Phys. 47, 609–644 (1987) 17. Collet, P., Koukiou, F.: Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329– 342 (1992) 18. Dodson, M.M., Rynne, B.P., Vickers, J.A.G.: Diophantine approximation and a lower bound for Hausdorff dimension. Mathematika 37, 59–73 (1990) 19. Dodson, M.M., Meli´an, M.V., Pestane, D., V´elani, S.L.: Patterson measure and Ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20(1), 37–60 (1995) 20. Falconer, K.J.: The multifractal spectrum of statistically self-similar measures. J. Theor. Prob. 7, 681–702 (1994) 21. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. New York: John Wiley, 1990 22. Falconer, K.J.: Representation of families of sets by measures, dimension spectra and Diophantine approximation. Math. Proc. Camb. Phil. Soc. 128, 111–121 (2000)
Multifractal Additive and Multiplicative Chaos
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23. Falconer, K.J.: One-sided multifractal analysis and points of non-differentiability of devil’s staircases. Math. Proc. Camb. Phil. Soc. 136, 167–174 (2004) 24. Fan, A.H.: Multifractal analysis of infinite products. J. Stat. Phys. 86, 1313–1336 (1997) 25. Feng, D.-J., Olivier, E.: Multifractal analysis of weak Gibbs measures and phase transition - application to some Bernoulli convolutions. Ergod. Th. Dynam. Sys. 23, 1751–1784 (2003) 26. Frisch, U., Parisi, G.: Fully developed turbulence and intermittency. Proc. International Summer School Phys., Enrico Fermi, Amsterdam: North Holland, 1985, pp. 84–88 27. Halsey, T., Jensen, M., Kadanoff, L., Procaccia, I., Shraiman B.: Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33(2), 1141–1151 (1986) 28. Harte, D.: Multifractals: Theory and Applications. Dordrecht: Kluwer/Chapman & Hall, 2001 29. Holley, R., Waymire, E.C.: Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab. 2, 819–845 (1992) 30. Jaffard, S.: Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3(1), 1–22 (1997) 31. Jaffard, S.: The multifractal nature of L´evy processes. Probab. Theory Relat. Fields 114(2), 207–227 (1999) 32. Jaffard, S.: On lacunary wavelet series. Ann. of Appl. Prob. 10(1), 313–329 (2000) 33. Kahane, J.-P.: Sur le chaos multiplicatif. Ann. Sci. Math. Qu´ebec 9, 105–150 (1985) 34. Kahane, J.-P., Peyri`ere, J.: Sur certaines martingales de Benoˆıt Mandelbrot. Adv. Math. 22, 131–145 (1976) 35. Khanin, K., Kifer,Y.: Thermodynamic formalism for random transformations and statistical mechanics. In: Sina˘ı’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Providence, RI: Amer. Math. Soc., 1996, pp. 107–140 36. Kifer, Y.: Fractals via random iterated function systems and random geometric constructions. In: Fractal geometry and stochastics (Finsterbergen, 1994) Progr. Probab. 37, Basel: Birkh¨auser,1995, pp. 145–164 37. Ledrappier, F., Porzio, A.: On the multifractal analysis of Bernoulli convolutions. I. Large-deviation results, II. Dimensions. J. Stat. Phys. 82, 367–420 (1996) 38. L´evy V´ehel, J., Riedi, R.H.: TCP traffic is multifractal: a numerical study, INRIA research report, RR-3129 (1997) 39. L´evy V´ehel, J., Vojak, R.: Multifractal analysis of Choquet capacities. Adv. Appl. Math. 20, 1–43 (1998) 40. Mandelbrot, B.: Intermittent turbulence in self-similar cascades: divergence of hight moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974) 41. Mandelbrot, B.: Fractals and Scaling in finance (Discontinuity, Concentration, Risk). Berlin-Heidelberg-New York: Springer, 1997 42. Mandelbrot, B., Evertsz, C.J., Hayakawa, Y.: Exactly self-similar left-sided multifractal measures. Phys. Rev. A 42(8), 4528–4536 (1990) 43. Molchan, G.M.: Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681–702 (1996) 44. Olsen, L.: Random Geometrically Graph Directed Self-similar Multifractals. Pitman Res. Notes Math. Ser. 307, 1994 45. Olsen, L.: A multifractal formalism. Adv. Math. 116, 92–195 (1995) 46. Parry, W., Policott, M.: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Soci´et´e Math´ematique de France, Ast´erisque 187–188 (1990) 47. Pollicott, M., Weiss, H.: Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation. Commun. Math. Phys. 207, 145–171 (1999) 48. Pesin, Y., Weiss, H.: The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1), 89–106 (1997) 49. Philipp, W., Stout, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2, 161 (1975) 50. Rand, D.A.: The singularity spectrum f (α) for cookie-cutters. Ergod. Th. Dynam. Sys. 9, 527–541 (1989) 51. Riedi, R.H., Mandelbrot, B.: Exceptions to the Multifractal Formalism for Discontinuous Measures. Math. Proc. Cambr. Phil. Soc. 123, 133–157 (1998) 52. Riedi, R.H.: Multifractal processes. In: Doukhan, Paul (ed.) et al., Theory and applications of longrange dependence, Boston: Birkh¨auser, 2003, pp. 625–716 53. Ruelle, D.: Statistical mechanics: Rigorous results. New York-Amsterdam: W. A. Benjamin, Inc., 1969 54. Takens, F., Verbitski, E.: General multifractal analysis of local entropies. Fund. Math. 165, 203–237 (2000) Communicated by J.L. Lebowitz
Commun. Math. Phys. 257, 499–514 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1299-4
Communications in
Mathematical Physics
A Local Quantum Version of the Kolmogorov Theorem David Borthwick1, , Sandro Graffi2 1
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA. E-mail: [email protected] 2 Dipartimento di Matematica, Universit`a di Bologna, 40127 Bologna, Italy. E-mail: [email protected] Received: 7 June 2004 / Accepted: 30 August 2004 Published online: 25 February 2005 – © Springer-Verlag 2005
Abstract: Consider in L2 (Rl ) the operator family H () := P0 (, ω) + Q0 . P0 is the quantum harmonic oscillator with diophantine frequency vector ω, Q0 a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and ∈ R. Then there exists ∗ > 0 with the property that if || < ∗ there is a diophantine frequency ω() such that all eigenvalues En (, ) of H () near 0 are given by the quantization formula Eα (, ) = E(, ) + ω(), α + |ω()|/2 + O(α)2 , where α is an l-multi-index. 1. Introduction and Statement of the Results Denote by Fρ,σ the set of all functions f (x, ξ ) : R2l → C with finite f ρ,σ norm for some ρ > 0, σ > 0 (see Sect. 2 for the definition and examples). Any f ∈ Fρ,σ is analytic on R2l and extends to a complex analytic function in the region |zi | ≤ ai | zi | for suitable ai > 0; moreover |f (z)| → 0 as |z| → +∞. Here z := (x, ξ ). Let ρ,σ denote the class of semiclassical Weyl pseudodifferential operators F in L2 (Rl ) with symbol f (x, ξ ) in Fρ,σ ; namely, (notation as in [Ro]) (1.1) (F u)(x) := OphW (f (x, ξ ))u(x) 1 = l ei(x−y),ξ / f ((x + y)/2, ξ )u(y) dydξ, u ∈ S(Rl ). Rl ×Rl It follows directly from the definition of f ρ,σ in (2.5) that F ∈ ρ,σ extends to a continuous operator in L2 (Rl ), with F L2 →L2 ≤ f ρ,σ . Consider in L2 (Rl ) the operator family H () = P0 (, ω) + Q0 and assume:
Supported in part by NSF grant DMS-0204985.
(1.2)
500
D. Borthwick, S. Graffi
(A1) P0 (, ω) is the harmonic-oscillator Schr¨odinger operator with frequencies ω ∈ [0, 1]l : 1 P0 (, ω)u = − 2 u + [ω12 x12 + . . . + ωl2 x12 ]u, 2 D(P0 ) = H 2 (Rl ) ∩ L22 (Rl ).
(1.3)
(A2) Q0 ∈ ρ,σ ; its symbol q0 (x, ξ ) = q0 (z) is real-valued for z = (x, ξ ) ∈ Rl × Rl , and q0 (z) = O(z2 ) as z → 0. (A3) There exist τ > l − 1, γ > 0 such that |ω, k| ≥ γ |k|−τ ,
∀k ∈ Zl \ {0},
|k| := |k1 | + . . . + |kl |, ω := (ω1 , . . . , ωl ). (1.4)
Denote 0 the set of all ω ∈ [0, 1]l fulfilling (1.4), and | 0 | its measure. It is well known that | 0 | = 1. Under the above assumptions the operator family H () defined on D(P0 ) is self-adjoint with pure-point spectrum ∀ ∈ R: Spec (H ()) = Specp (H ()). Moreover (1.4) entails in particular the rational independence of the components of ω and hence the simplicity of Spec(P0 ) and its density in R+ := R+ ∪ {0}. Clearly, P0 is a semiclassical pseudodifferential operator of order 2 with symbol 1 (|ξ |2 + |ωx|2 ) 2 l 1 1 = ωk Ik (x, ξ ), Ik (x, ξ ) := [ξ 2 + ωk2 xk2 ], k = 1, . . . , . (1.5) 2 2ωk k
p0 (x, ξ ) =
k=1
Theorem 1.1. Let (A1-A3) be verified; let h∗ > 0. Then given η > 0 there exist ∗ > 0 and, for all ∈ [− ∗ , ∗ ], ⊂ 0 independent of ( ∈ [0, ∗ ], η) and ω(, ) ∈ , such that if |α| < η the spectrum of H () is given by the quantization formula 1 Eα (, ) = E(; ) + ω(, ), α + |ω(, )| + R(α, ; ). 2
(1.6)
Here: 1. E(x; ) : [0, h∗ ]×[− ∗ , ∗ ] → R is continuous in x and analytic in , with E(x, 0) = 0, E(0; ) = 0; 2. ω(x; ) : [0, h∗ ] × [− ∗ , ∗ ] → R is continuous in x and analytic in with ω(x; 0) = ω. l 3. R(x, y, ) : R+ × [0, h∗ ] × [− ∗ , ∗ ] → R is continuous in (x, y; ) and such that |R(x, y; )| = O(|x|2 ),
(1.7)
uniformly with respect to (y, ). 4. | − 0 | → 0 as → 0. The uniformity in of the estimates needed to prove Theorem 1.1 yields in this particular setting a formulation of Kolmogorov’s theorem equivalent to that of [BGGS]:
A Local Quantum Version of the Kolmogorov Theorem
501
Corollary 1.1. Let ∗ , , E(x; ), ω(x; ) be as above. Then ∀ there is an analytic l canonical transformation (x, ξ ) = ψ (I, φ) of R2l onto R+ × Tl such that ˜ φ; ). (p ◦ ψ)(I, φ) = E() + ω(), I + R(I,
(1.8)
Here p (x, ξ ) is the symbol of H (), E() := E(0; ), ω() := ω(0; ) ∈ ; ˜ R(I, φ; ) = O(I 2 ) as I → 0 uniformly in φ. Remarks. 1. The form (1.8) of the Hamiltonian entails that a quasi periodic-motion with diophantine perturbed frequency ω() ∈ exists on the unperturbed torus I = 0; equivalently, a quasi periodic motion with frequency ω() ∈ exists on the perturbed torus with parametric equations (x, ξ ) = ψ (0, φ). Making I = α (1.6) represents the quantization of the r.h.s. of (1.8). In the formulation of [BGGS] a quasi periodic motion with the unperturbed frequency ω ∈ exists on a perturbed torus with parametric equations (x, ξ ) = ψ (0, φ). The selection of the diophantine frequency within depends here on because of the isochrony of the Hamiltonian flow generated by p0 . 2. KAM theory (see e.g. [Ko, AA, Mo]) was first introduced in quantum mechanics in [DS] to deal with quasi-periodic Schr¨odinger operators. For its applications to the Floquet spectrum of non-autonomous Schr¨odinger operators see [BG] and references therein. Its first application to generate quantization formulas for fixed goes back to [Be] for operators in L2 (Tl ) and to [Co] for non-autonomous perturbations of the harmonic oscillators. If quasi-modes are considered instead of eigenvalues, a uniform quantum version of the Arnold version has been obtained by Popov [Po2], within a quantization different from the canonical one [CdV]. 3. The related method of the quantum normal forms also yields quantization formulas with remainders, valid for a much more general class of perturbation of the harmonic oscillators, which in particular includes the polynomials in x; however they apply only to perturbations of semi-excited levels ([Sj, BGP]) or again require quasi-modes and a quantization different from the canonical one [Po1]. More precisely, in [Sj] a quantization formula with remainder of order O(∞ ) has been constructed for all eigenvalues lying in the interval [0, δ ], δ > 0 ("semi-excited levels"). This result has been sharpened in [BGP]: an estimate of Nekhoroshev type uniform with respect to holds for the remainder of the quantum normal form when the perturbation fulfills A1)-A3) above; appropriate approximation and scaling arguments turn the O(∞ ) a remainder estimate of [Sj] into the quantitative estimate O(e−1/ ), 0 < a < 1 for a slightly more general class of semi-excited levels. For quasi-modes these results can be further sharpened to an error estimate of the type O(e−1/ ) valid also for excited levels [Po1]. 2. Proof of the Results Define an analytic action of Tl into R2l through the flow of p0 : : Tl × R2l → R2l ,
φ, (x, ξ ) → (x , ξ ) = φ (x, ξ ),
(2.1)
where xk :=
ξk sin φk + xk cos φk , ξk := ξk cos φk − ωk xk sin φk . ωk
(2.2)
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D. Borthwick, S. Graffi
If z := (x, ξ ), the flow of initial datum z0 is indeed z(t) = ωt (z0 ), ωt := (ω1 t, . . . , ωl t). If f ∈ L1loc (R2l ), the angular Fourier coefficient of order k is defined by 1 f (φ (z))e−ik,φ dφ, k ∈ Zl . f˜k (z) := (2π)l Tl If f ∈ C1 one has, as is well known, f˜k (z)eik,φ ⇒ f (z) = f˜k (z). f (φ (z)) = k∈Zl
k∈Zl
Note furthermore that f ≡ f˜k for some fixed k if and only if f (φ (z)) = eik,φ f (z). Taking f ∈ L1 (R2l ), we will consider the space Fourier transform 1 f (z)e−is,z dz, f(s) := (2π)2l R2l
(2.3)
(2.4)
as well the space Fourier transforms of the f˜k ’s: 1 f (φ (z))e−ik,φ e−is,z dφ dz. f˜k (s) := (2π)3l R2l Tl Given ρ > 0, σ > 0, define the norm eρ|k| f ρ,σ := k∈Zl
R2l
|f˜k (s)|eσ |s| ds.
(2.5)
Definition 2.1. Let ρ > 0, σ > 0. Then Fρ,σ := {f : R2l → C | f ρ,σ < +∞}. Remarks. 1. If f ∈ Fρ,σ then f is analytic on R2l , and extends to a complex analytic function on a region Bρ,σ ⊂ C2l of the form Bρ,σ := |zi | ≤ ai | zi |, with suitable ai . 2. F :=OphW (f ) is a trace-class, and, for real-valued f , self-adjoint -pseudodifferential operator in L2 (Rl ) if f ∈ Fρ,σ . Let f(s) be the Fourier transform of f . Since fL1 ≤ f ρ,σ , we have F L2 →L2 ≤ |f(s)| ds ≡ fL1 , F L2 →L2 ≤ f ρ,σ . (2.6) R2l
3. We introduce also the space Fσ of all functions f : R2l → C such that | g (s)|eσ |s| ds < +∞. gσ := R2l
Obviously if f ∈ Fσ then f is analytic on R2l , and extends to a complex analytic function in the multi-strip S := {z ∈ C 2l | |zi | < σ }. 2 2 4. Example of f ∈ Fρ,σ : f (x, ξ ) = P (x, ξ )e−(|x| +|ξ | ) , P (x, ξ ) any polynomial. The starting point of the proof is represented by the first step of the Kolmogorov iteration, and is summarized in the following
A Local Quantum Version of the Kolmogorov Theorem
503
Proposition 2.1. Let ω ∈ 0 . Then, for any 0 < d < ρ, 0 < δ < σ : 1. There exists a unitary transformation U (ω, , ) = eiW1 / : L2 ↔ L2 , W1 = W1∗ and ω1 () ∈ [0, 1]l such that: U H ()U −1 = P0 (, ω1 ()) + E1 Id + 2 Q1 (, ) + R1 (, ).
(2.7)
Here: E1 = q˜0 ; Id is the identity operator; W1 = OphW (w1 ) ∈ ρ−d,σ −δ , Q1 (, ) = OphW (q1 ) ∈ ρ−d,σ −δ with w1 ρ−d,σ −δ ≤ d −τ q0 ρ,σ
q1 ρ−d,σ −δ ≤ δ −2 d −2τ q0 2ρ,σ .
(2.8)
2. R1 () is a self-adjoint semiclassical pseudodifferential operator of order 4 such that [R1 (), P0 ] = 0; ∃ D1 > 0 such that, for any eigenvector ψα of P0 (ω): |ψα , R1 ()ψα | ≤ D1 (|α|)2 . (2.9) γ 3. ∀ K > 0 with (1 + K τ ) < ∃ 1 ⊂ 0 closed and d1 > 1 independent of q0 ρ,σ K such that 1
| 0 − 1 | ≤ γ (1 + 1/K d ).
(2.10)
Moreover if ω1 ∈ 1 then (1.4) holds with γ replaced by γ1 := γ − q0 ρ,σ (1 + K τ ).
(2.11)
Proof. To prove Assertion 1 we first recall some relevant results of [BGP]. Lemma 2.1 (Lemma 3.6 of [BGP]). Let g ∈ Fρ,σ . Then the homological equation, {p0 , w} + N = g,
{p0 , N } = 0
(2.12)
admits the analytic solutions N := g˜ 0 ;
w :=
k=0
g˜ k , iω, k
(2.13)
with the property N ◦ φ = N . Equivalently, N depends only on I1 , . . . , Il . Moreover, for any d < ρ: τ τ 1 gρ,σ N ρ,σ ≤ gρ,σ ; wρ−d,σ ≤ c ; c := . (2.14) dτ e γ Given (g, g ) ∈ Fρ,σ , let {g, g }M be their Moyal bracket, defined as {g, g }M = g#g − g #g, where # is the composition of g, g considered as Weyl symbols. We recall that in Fourier transform representation, used throughout the paper, the Moyal bracket is (see e.g. [Fo],§3.4): 2 ({g, g }M )∧ (s) = g (s 1 )g (s − s 1 ) sin (s − s 1 ) ∧ s 1 /2 ds 1 , (2.15) R2n where, given two vectors s = (v, w) and s 1 = (v 1 , w1 ), s ∧ s 1 := w, v1 − v, w1 . We also recall that {g, g }M = {g, g } if either g or g is quadratic in (x, ξ ).
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Lemma 2.2 (Lemmas 3.1 and 3.4 of [BGP]). Let g ∈ Fσ , g ∈ Fσ −δ . Then: 1. ∀ 0 < δ < σ − δ: {g, g }M σ −δ−δ ≤
1 e2 δ (δ
+ δ)
gσ g σ −δ .
(2.16)
2. Let g ∈ Fρ,σ and g ∈ Fρ,σ −δ . Then, for any positive δ < σ − δ: {g, g }M ρ,σ −δ−δ ≤
1 gρ,σ g ρ,σ −δ . e2 δ (δ + δ )
(2.17)
As a simple corollary of Lemmas 2.1 and 2.2, we find: Lemma 2.3 (Lemma 3.5 and Lemma 3.7 of [BGP]). Let g ∈ Fρ,σ , w ∈ Fρ,σ . 1. Define gr :=
1 {w, gr−1 }M , r
r ≥ 1; g0 := g.
Then gr ∈ Fρ,σ −δ for any 0 < δ < σ , and the following estimate holds: r gr ρ,σ −δ ≤ δ −2 wρ,σ gρ,σ .
(2.18)
2. Let g ∈ Fρ,σ , and w be the solution of the homological equation (2.12). Define the sequence pr0 : r = 0, 1, . . . as follows: p00 := p0 ;
pr0 :=
1 {w, pr−10 }M , r ≥ 1. r
Then, for any 0 < d < ρ, 0 < δ < σ , pr0 ∈ Fρ−d,σ −δ and fulfills the following estimate: r−1 gρ−d,σ , r ≥ 1. pr0 ρ−d,σ −δ ≤ 2 δ −2 wρ−d,σ Proof of Proposition 2.1. With U1 = eiW1 / , W1 continuous and self-adjoint, we have in general: U1 (P0 + Q0 )U1−1 = P0 + P1 + 2 Q1 ,
(2.19)
P1 := Q0 + [W1 , P0 ]/ i,
(2.20)
Q1 := −2 U1 (P0 + Q0 )U1−1 − P0 − (Q0 + [W1 , P0 ]/ i) .
(2.21)
We start by looking for W1 ∈ Fρ,σ such that the first order term yields an operator N1 ∈ Fρ,σ commuting with P0 : Q0 + [W1 , P0 ]/ i = N1 ,
[N1 , P0 ] = 0.
(2.22)
Denoting by w1 , N1 the (Weyl) semiclassical symbols of W1 , N1 , respectively, Eq.(2.22) is equivalent to a classical homological equation in Fρ,σ , {p0 , w1 }M + N1 = q0 ,
{p0 , N1 }M = 0.
(2.23)
A Local Quantum Version of the Kolmogorov Theorem
505
However p0 is quadratic in (x, ξ ). Therefore the Moyal bracket {p0 , w1 }M coincides with the Poisson bracket {p0 , w1 } and the above equation becomes {p0 , w1 } + N1 = q0 ,
{p0 , N1 } = 0.
(2.24)
The existence of w1 ∈ Fρ−d,σ , N1 ∈ Fρ,σ with the stated properties now follows by direct application of Lemma 2.1. We now prove the second estimate in (2.8). We have: 1 s Q1 = eis1 W1 / [[P0 + Q0 , W1 ], W1 ]e−is1 W1 / ds1 ds, 0
0
and we can estimate [[P0 + Q0 , W1 ], W1 ]L2 →L2 ≤ {{p0 + q0 , w1 }M , w1 }M ρ−d,σ −δ . It follows, by Lemma 2.3 and Lemma 2.1, that Q1 L2 →L2 ≤ {{p0 + q0 , w1 }M , w1 }M ρ−d,σ −δ ≤ δ −2 d −2τ q0 2ρ,σ . This proves the second estimate of (2.8). To prove Assertion 2 set: E1 := N1 (0); ω1 () = ω + (∇I N1 )(0), R1 (I, ) = N1 (I ) − (∇I N1 )(0), I − E1 ,
(2.25) (2.26)
R1 () := OphW (R1 (I, )).
(2.27)
and define
Then clearly R1 () is a self-adjoint semiclassical, tempered pseudodifferential operator of order 4, vanishing to 4th order at the origin, and with the property [R1 (), P0 ] = 0. Therefore formula (2.9) follows directly by Proposition A.1. As far as Assertion 3 is concerned, set: Tk (α) := {ω ∈ [0, 1]k : |ω, k| ≤ α},
γ1 1 := 0 − . Tk |k|τ
(2.28) (2.29)
|k|≥K
As in [BG], Lemma 5.6, we have: |Tl (α)| ≤
4 α. k
Hence if τ > l − 1 we can write
γ1 γ1 γ1 ≤ Tk < d . τ τ +1 1 |k| |k| K |k|≥K |k|≥K Since |ω1 (), k| ≥ γ1 /|k|τ by construction when |k| ≤ K, the proposition is proved.
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3. Iteration The above result represents the starting point for the iteration. To ensure convergence, we first preassign the values of the parameters involved in the iterative estimates. Keeping , K, γ , ρ and σ fixed define, for p ≥ 1: σp :=
σ , 4p 2
sp := sp−1 − σp ,
γp := γp−1 −
4p , 1 + Kpτ
ρp :=
ρ , 4p 2
rp := rp−1 − ρp ,
(3.1)
Kp := pK,
(3.2)
where p is defined in (3.15) below. The initial values of the parameter sequences are chosen as follows: γ0 := γ ;
s0 := σ ;
0 = 0.
r0 := ρ,
(3.3)
We then have: Proposition 3.1. Let ω ∈ 0 . There exist ∗ (γ ) > 0 and, ∀ p ≥ 1, a closed set p ⊂ 0 γ such that, if || < ∗ (γ ) > 0 and ωp (; ) ∈ p : γ
1. One can construct two sequences of unitary transformations {Xp }, {Yp } in L2 (Rl ) with the property Xp (P0 (ω) + Q0 )Xp−1 = P0 (ωp (; )) p
+ Ep (; )Id + e2 Qp + 2 +
p
Ys Rs−1 ()Ys−1 2
s−2
p−1
Rp (; )
.
(3.4)
s=2
2. Xp and Yp have the form Xp = Up Up−1 · · · U1 , Ys = Up Up−1 · · · Us . Here Up (ω, , ) = exp [i 2
p−1
(3.5) (3.6)
Wp /] : L2 ↔ L2 , Wp = Wp∗ ,
Wp = OphW (wp ) ∈ rp ,sp , wp rp ,sp ≤ ρp−2τ qp−1 rp−1 ,sp−1 , Ep (; ) =
Qp (, ) = OpW (qp ) ∈ rp ,sp ,(3.7)
qp rp ,sp ≤ ρp−2τ σp−2 qp−1 2rp−1 ,sp−1 ,(3.8) p
s
Ns () 2 ,
Ns () = (q˜s )0 ().
(3.9)
s=0
3. Rs () is a self-adjoint semiclassical pseudodifferential operator of order 4; [Rs (), P0] = 0; there exist Dp > 0, D p > 0 such that, for any eigenvector ψα of P0 (ω): |ψα , Rp ()ψα | ≤ Dp (|α|)2 , |ψα ,
p s=2
Ys Rs−1 Ys−1 2
s−2
ψα | ≤ D p (|α|)2 .
(3.10) (3.11)
A Local Quantum Version of the Kolmogorov Theorem
507
4. ∀ Kp−1 > 0 such that τ (1 + Kp−1 )<
γp−1 , qp−1 rp−1 ,sp−1
(3.12)
∃ p ⊂ p−1 closed and dp > 1 independent of Kp such that | p − p−1 | ≤
γp−1 . 1 + 1/(Kp−1 )dp
(3.13)
Moreover if ωp () ∈ p then (1.4) holds with γ replaced by τ ), γp := γp−1 − p (1 + Kp−1
p :=
2p−1
(3.14)
qp−1 rp−1 ,sp−1 .
(3.15)
Proof. We proceed by induction. For p = 1 the assertion is true because we can take W1 , Q1 , R1 , ω1 , 1 , K1 as in Proposition 2.1. To go from step p − 1 to step p we consider the operator −1 := P0 (ωp−1 (; )) + Ep−1 (; )I + 2 Xp−1 (P0 (ω) + Q0 )Xp−1
+ 2
p−1
Rp−1 (; ) +
p−1
p−1
Qp−1
Ys Rs−1 ()Ys−1 2
s−2
.
s=2
We have to determine and estimate the unitary map Up transforming it into the form (3.4) via the definitions (3.5). With Up = ei we have at the pth iteration step Up (P0 (ωp−1 + 2
p−1
2p−1 W / p
, Wp continuous and self-adjoint,
Qp−1 )Up−1 = P0 (ωp ) + 2 Pp + 2 Qp , Pp := Qp−1 + [Wp , P0 ]/ i, p−1
p
Qp := −2 Up (P0 (ωp−1 ) + Q0 )U1−1 − P0 (ωp−1 ) − (Qp−1 + [Wp , P0 ]/ i) (the explicit dependence of the frequencies on (, ) has been omitted). We will look therefore for Wp ∈ rp ,sp and an operator Np ∈ rp ,sp such that Qp + [Wp , P0 ]/ i = Np ,
[Np , P0 ] = 0.
(3.16)
Denoting wp , Np the (Weyl) semiclassical symbols of Wp , Np , respectively, Eq. (3.16) is again equivalent to the classical homological equation in Fρ,σ , {p0 , wp }M + Np = qp ,
{p0 , Np }M = 0,
which once more becomes {p0 , wp } + Np = qp ,
{p0 , Np } = 0.
The existence of wp ∈ Frp ,sp , Np ∈ Frp ,sp with the stated properties now follows by direct application of Lemma 2.1. Expanding Np as in the proof of Proposition 2.1 and taking into account the definitions (3.5) we immediately check that Up Xp−1 (P0 (ω) +
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D. Borthwick, S. Graffi
−1 Q0 )Xp−1 Up−1 has the form (3.4). The estimate of Qp and the small denominator estimates follow by exactly the same argument of Proposition 2.1. The estimate (3.10) is proved exactly as (2.9). It remains to prove the estimate (3.11). By the inductive assumption, it is enough to prove the existence of Dp > 0 such that
|ψα , Up Rp−1 Up−1 ψα | ≤ Dp (|α|)2 . We only have to prove that the operator Up Rp−1 Up−1 is an -pseudodifferential operator of order 4 fulfilling the hypotheses of Proposition A.1, assuming by the inductive argument the validity of these properties for Rp−1 . On the other hand, Up = p−1 exp (i 2 Wp /), and Wp is an -pseudodifferential operator of order 0. We can therefore apply the semiclassical Egorov theorem (see e.g. [Ro], Chapter 4) to assert that Up Rp−1 Up−1 is again an -pseudodifferential operator. Denote σ (x, ξ ; ; ) the Weyl symbol of Up Rp−1 Up−1 , and consider its expansion σ (x, ξ ; ; ) = σ0 (x, ξ ; ) +
M
j σj (x, ξ ; ) + O(hM+1 ).
j =1
It is clearly enough to prove that the principal symbol σ0 (x, ξ ; ) has order 4. Denote by p
φ(x, ξ ; ) := exp [ 2 Lwp ](x, ξ ) p
the Hamiltonian flow on R2l generated by the Hamiltonian vector field Lwp at time 2 ; here wp0 (x, ξ ) is the principal symbol of Wp . Then σ0 (x, ξ ; ) = R0p−1 (φ(x, ξ ; )), where R0p−1 (x, ξ ) is in turn the principal symbol of Rp−1 . Now
2
φ(x, ξ ; ) = (x +
p
∇ξ wp (x, ξ ; η) dη, ξ −
0
2
p
∇x wp (x, ξ ; η) dη).
0
By Assumption A2 and the inductive hypothesis we know that wp (z) = O(|z|2 ) as |z| → 0. Hence we can write φ(z) = z + r(z), where r(z) = O(z), z → 0. This concludes the proof of Proposition 3.1. Proof of Theorem 1.1. Applying the estimates on qp in Propositions 2.1 and 3.1 iteratively, we have 2 2τp 2 2p 4p 4p p qp rp ,sp ≤ · q0 2 , (3.17) ρ σ whence ||2 Qp L2 →L2 ≤ ||2 (4p 2 )2p(τ +1) ρ −2τp σ −2p q0 2 → 0 p
p
p
as p → ∞, (3.18)
for all || ≤ ∗ provided ∗ > 0 is small enough. At the p th iteration the frequency is given by ωp (; ) = ω +
p
∇I Ns () 2
s−1
s=1
Since ∇z f (z)ρ−d,σ −δ ≤
1 f (z)ρ,σ , by (3.17) we have dδ
.
(3.19)
A Local Quantum Version of the Kolmogorov Theorem p
s
|∇I Ns () 2 | ≤
s=1
p
509
||2 (4s 2 )2s(τ +1) ρ −2τ s σ −2s q0 2 . s
s
(3.20)
s=1
Hence the series (3.19) converges as p → ∞ for || < ∗ if ∗ is small enough, uniformly with respect to ) ∈ [0, h∗ ]. In the same way, the estimate (3.17) entails, by the definition (3.14), the existence of lim γp := γ∞ . Let ω(; ) := limp→∞ ωp ( ). p→∞
Then ω(; ) is diophantine with constant γ∞ by Proposition 3.1. In the same way: E(; ) =
∞
s
Ns () 2 ,
|| < ∗ .
s=1
Finally, let R(α, ) be an asymptotic sum of the power series
∞
Ys Rs−1 Ys−1 2
s−2
.
s=2
Then the validity of (1.7) follows by its validity term by term. This concludes the proof of Theorem 1.1. Proof of Corollary 1.1. It is enough to illustrate the specialization of the argument of Propositions 2.1 and 3.1 to the = 0 case. Denoting by e Lw1 the canonical flow at time generated by the Hamiltonian vector field generated by the symbol w1 , we have: e Lw1 (p0 + q0 )(x, ξ ) = (p0 + p1 + 2 q10 )(x, ξ ), p1 := q0 + {w1 , p0 }, 0 −2 Lw1 q1 := (p0 + q0 )(x, ξ ) − p0 − (q0 + {w1 , p0 }) . e
(3.21) (3.22) (3.23)
Remark that e Lw1 (p0 + q0 )(x, ξ ) is the principal symbol of U1 (P0 + Q0 )U1−1 by the semiclassical Egorov theorem; p1 is the full, and hence principal, symbol of P1 because p0 is quadratic. Likewise, q10 is the principal symbol of Q1 . Hence the classical definitions (3.21,3.22,3.23) correspond to the principal symbols of the semiclassical pseudodifferential operators U1 (P0 + Q0 )U1−1 , P1 , Q1 defined in (2.19,2.20,2.21). Therefore we can take over the homological equation (2.24) and apply Lemma 2.1 once more. This yields the same w1 and N1 of Proposition 2.1. To prove the estimate (2.8) for q10 we write 1 0 q1 = es Lw1 {{p0 + q0 , w1 }, w1 } ds. 0
Now as in [BGGS], Lemma 1, note that if || < ∗ and z = (x, ξ ) ∈ Bρ−d,σ −δ then es Lw1 z ∈ Bρ,σ for 0 ≤ s ≤ 1 because (Lemma 2.1) ∇w1 ρ−d,σ ≤ (τ/e)cψ d −τ q0 ρ,σ . Therefore we can apply Lemma 2.3, valid a fortiori for the Poisson bracket, and, as in the proof of Proposition 2.1, get the estimate corresponding to the second one of (2.8): q10 ρ−d,σ −δ ≤ {{p0 + q0 , w1 }, w1 }ρ−d,σ −δ ≤ δ −2 d −2τ q0 2ρ,σ .
(3.24)
Now, writing: ψ1 (x, ξ ) = e Lw1 (x, ξ ), E1 := N1 (0), ω1 () = ω + (∇I N1 )(0), ˜ 1 (I, ) = N1 (0) − (∇I N1 )(0), I − E1 , R
(3.25) (3.26) (3.27)
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D. Borthwick, S. Graffi
we can sum up the above argument by writing (compare with (2.7)) ψ1 ◦ (p0 + q0 ) = E1 + ω1 (0; ), I + 2 q1 (I, φ) + R01 (I, ),
(3.28)
where R01 is the principal symbol of R1 . Morover, Assertion 3 of Proposition 2.1 holds without change. Let us now specialize the iterative argument of Proposition 3.1. First, the parameters defined in (3.1,3.2,3.3) remain unchanged. Then: 1. The construction of the two sequences of canonical transformations χp = ψ1 ◦ ψ2 · · · ◦ ψp , ζs
=
p = 1, 2, . . . ,
ψp ◦ ψp−1 · · · ◦ ψs , Lw0
ψs (x, ξ ) = e
(3.29)
p = 1, 2, . . . ,
(3.30)
(x, ξ ),
(3.31)
ψ,I0 ◦ (p0 + q0 ) =
(3.32)
s
such that p
p
p
ωp (0, ), I + Ep () + e2 qp0 + 2 R0p +
p
ψs ◦ R0s−1 2
s−2
s=2
follows as in the above argument valid for p = 1. Here ws0 , qp0 , R0s are the principal symbols of the semiclassical pseudodifferential operators Ws , Qp and Rs , once reexpressed on the (x, ξ ) canonical variables via, with ωp in place of ω1 . Moreover: Ep () =
p
s
Ns (0) 2 ,
Ns (0) = (q˜s0 )0 (0),
(3.33)
s=0
ωp () = ω +
p
s
ωs (0) 2 ,
ωs (0) = ∇I Ns (0).
(3.34)
s=0
2. The estimates (3.8) are a fortiori valid with wp0 , qp0 in place of wp , wp ; as a consequence, (3.13) holds unchanged together with the definitions (3.12,3.14,3.15). Hence the uniform estimate (3.17) allows us to set = 0 in (3.19,3.20). 3. Finally, remark that R0s (I ) = O(I 2 ), s = 1, . . . , p. Now the estimate ψs Rs (I ) = O(I 2 ) as I → 0 follows by exactly the same argument of Proposition 3.1 after re-expression on the canonical variables (x, ξ ). Appendix To establish the remainder estimate (1.7) the key fact is that vanishing of a symbol at the origin (x, ξ ) = 0 implies bounds on harmonic oscillator matrix elements that are uniform in . No analyticity of the symbol is required for this result, so we will state and prove it in somewhat greater generality, using the following semiclassical symbol class defined in Shubin [Sh]: m,µ = {f ∈ C ∞ (R2l × (0, ]) : |∂z f (z, )| ≤ Cγ zm−|γ | µ }, γ
A Local Quantum Version of the Kolmogorov Theorem
where z = (x, ξ ), here considered a real variable, and z = reference we note that Proposition A.2.3 of [Sh] gives the result: ∀f ∈ 0,µ ,
511
1 + |z|2 . For future
OpW (f )L2 ≤ C(f )µ ,
(A.1)
for all ∈ (0, ]. The matrix elements in question are most easily computed in Bargmann space, with the remainder operator written as a Toeplitz operator. Since these are anti-Wick ordered, we first must consider the translation from Weyl symbols to anti-Wick (for these notions, see e.g. [BS]). Denoting by OpAW (f ) the anti-Wick quantization of a symbol f ∈ m,µ , the correspondence is given by the action of the heat kernel on the symbol: OpAW (f ) = OpW (e /4 f ),
(A.2)
where = z = ∂x · ∂x + ∂ξ · ∂ξ . To begin, we show that the Weyl symbol of an anti-Wick operator is given by formal expansion of the heat kernel up to a remainder. Lemma A.1. For f, g ∈ m,µ , suppose that OpAW (g) = OpW (f ). Then for all n ≥ 1,
k n−1 1 g ∈ m−2n,µ+n . f− k! 4 k=0
Proof. According to (A.2), f (z, ) =
1 (π)l
2 e−|z−w| / g(w)dw.
In this expression we will expand g(w) in a Taylor series centered at w = z: 1 ∂ α g(z, )(w − z)α + r(w, z, ), α!
g(w, ) =
|α|<2n
where r(w, z) =
cα (w − z)α
|α|=2n
Thus, f (z, ) =
1
(1 − t)2n−1 ∂ α g(z + t (w − z)) dt.
0
cα ∂ α g(z, ) + r(z, ),
|α|<2n
where cα = and r(z, ) =
|α|=2n
cα −l
1
1 1 (π)l α!
2 wα e−|w| / dw,
2 (w − z)α e−|z−w| / (1 − t)2n−1 ∂ α g(z + t (w − z)) dt dw.
0
Note that cα = 0 for |α| odd, and for any integer k,
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D. Borthwick, S. Graffi
1 cα ∂ g = k!
α
|α|=2k
k
4
g.
The lemma is thus reduced to the claim that r(z, ) ∈ √ m−2n,µ+n . To see this, we change variables by w = (w − z)/ to write 1 √ 2 n r(z, ) = cα w α e−|w| (1 − t)2n−1 ∂ α g(z + tw ) dt dw. 0
|α|=2n
We must estimate the derivatives: 1 √ 2 ∂ γ r(z, ) = cα n w α e−|w| (1 − t)2n−1 ∂ β g(z + tw ) dt dw, 0
|α|=2n
where |β| = 2n + |γ |. This integral for ∂ γ r we then split into two pieces according to : |w| > |z|/2. The assumption the domain of the w-integral, Iα,β : |w| < |z|/2 and Iα,β m,µ g∈ implies an estimate | ≤ Czm−2n−|γ | n+µ . |Iα,β
(A.3)
The second term is taken care of by the exponential factor in |w|: | < Cl l z−l , |Iα,β
∀l.
Therefore ∂ γ r satisfies an estimate of the form (A.3) for any γ , and hence r ∈ m−2n,µ+n . Our application of Lemma A.1 will be specifically to operators of order 4: Lemma A.2. For g ∈ 4,0 , OpW (g) = OpAW (g) −
OpAW ( g) + R(), 4
where R()L2 ≤ C2 . Proof. Let σ (A) denote the Weyl symbol of the -pseudodifferential operator A. Applying Lemma A.1 with n = 2 gives σ (OpAW (g)) = g +
g + r1 , 4
and σ (OpAW ( g)) = g + r2 , 4 4 where r1 , r2 ∈ 0,2 . Noting that OpW (g) − OpAW (g) +
σ (OpAW ( g)) = OpW (r1 − r2 ), 4
the bound on R() follows from (A.1).
A Local Quantum Version of the Kolmogorov Theorem
513
The point of introducing anti-Wick symbols is to exploit the Bargmann space representation of the harmonic oscillator. The Bargmann space is (see e.g. [BS]) 2 H = L2hol (Cl , e−|z| / dzd z¯ ).
The Bargmann transform is an isomorphism B : L2 (Rl ) → H , defined so as to intertwine anti-Wick operators with Toeplitz operators: B ◦ OpAW (f ) ◦ B −1 = T (f ). The Toeplitz operator T (f ) : H → H is defined for f ∈ m,µ by T (f ) = M(f ), where M(f ) denotes the multiplication operator on L2 (Cl , e−|z| / dzd z¯ ) (identifying 2 R2l = Cl by z = x +iξ ), and : L2 (Cl , e−|z| / dzd z¯ ) → H is orthogonal projection onto the holomorphic subspace. The main result of this Appendix is the following matrix element estimate: 2
Proposition A.1. Let {ψα } be the normalized eigenstates of the standard harmonic oscillator on L2 (Rl ). Suppose f ∈ 4,0 satisfies f (z, ) = zγ gγ (z, ), |γ |=4
where
sup |∂ β g
γ|
≤ M for all |β| ≤ 2. Then |ψα , OpW (f )ψα | ≤ CM(|α|)2
for all α, , where C depends only on the dimension. Proof. Under the Bargmann transform the harmonic oscillator eigenstates have a particularly convenient form: (B −1 ψα )(z) = (π l |α|+l α!)−1/2 · zα . Using Lemma A.1 we write OpW (f ) = OpAW (f ) −
OpAW ( f ) + R(), 4
(A.4)
where |R()| ≤ C2 . Consider the matrix element of the first term on the right-hand side of (A.4). In Bargmann space this becomes 1 2 AW ψα , Op (f )ψα = l |α|+l z¯ α f (z, )zα e−|z| / dzd z¯ . π α! Writing f as a sum over zγ gγ with |γ | = 4, the estimate for a particular γ is straightforward: α 2 4 −|z|2 / 1 z |z| e |ψα , OpAW (zγ gγ )ψα | ≤ M l |α|+l dzd z¯ π α! = M2 (|α| + l)(|α| + l + 1). The second term
on the right in (A.4) is handled in a similar way. By assumption we can write f = |η|=2 zη hη (z, ), where sup |hη | ≤ 12M. The estimate then proceeds exactly as above (noting that there is an extra factor of in front of this term).
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Acknowledgement. We thank Dario Bambusi for many useful comments and for pointing out an error in the first draft of this paper.
References [AA]
Arnold, V. I., Avez, A.: Probl`emes ergodiques de la m`ecanique classique. Paris:Gauthier-Villars, 1967 [BG] Bambusi, D., Graffi, S.: Nonautonomous Schr¨odinger operators with unbounded quasiperiodic coefficients and KAM methods. Commun. Math. Phys. 219, 465–480 (2001) [BGP] Bambusi, D., Graffi, S., Paul, T.: Normal Forms and Quantization Formulae. Commun. Math. Phys. 207, 173–195 (1999) [BGGS] Benettin, G., Galgani, L., Giorgilli, A., Strelczyn, J. M.: A proof of Kolmogorov’s theorem. Nuovo Cimento 79B, 201–223 (1984) [BS] Berezin, F. A., Shubin, M. S.: The Schr¨odinger Equation. Dordrecht:Kluwer, 1991 [Be] Bellissard, J.: Stability and Instability in Quantum Mechanics. In: Trends and developments in the eighties (Bielefeld, 1982/1983), Singapore:World Sci. Publishing, 1985, pp. 1–106 [CdV] Colin de Verdi`ere, Y.: Quasimodes sur les variet´es Riemanniennes. Invent. Math. 43, 15–52 (1977) [Co] Combescure, M.: The quantum stability problem for time-periodic perturbations of the harmonic oscillator. Ann. Inst. H. Poincar´e Phys. Th´eor. 47(1), 63–83 (1987) [DS] Dinaburg, E. I., Sinai, Ya. G.: The one-dimensional Schr¨odinger operator with a quasi-periodic potential. Functional Anal.Appl. 9, 279–289 (1976) [Fo] Folland, G.: Harmonic analysis in phase space. Princeton, NJ:Princeton University Press, 1988 [Ko] Kolmogorov, A. N.: On conservation of conditionally periodic motions for a small change in Hamilton’s function (Russian). Dokl. Akad. Nauk SSSR (N.S.) 98, 527–530 (1954) (English translation in: G.Casati, J.Ford (eds.): Lecture Notes in Physics 91, Berlin-Heidelberg-New York:Springer-Verlag, 1979 [Mo] Moser, J.: Stable and random motions in Hamiltonian systems. Princeton, NJ:Princeton University Press, 1973 [Po1] Popov, G.: Invariant tori effective stability and quasimodes with exponentially small terms. I: Birkhoff normal form. Ann. Henri Poincar´e 1, 223–248 (2000) [Po2] Popov, G.: Invariant tori effective stability and quasimodes with exponentially small terms. II: Quantum Birkhoff normal form. Ann. Henri Poincar´e 1, 249–279 (2000) [Ro] Robert, D.: Autour de l’approximation semiclassique, Basel:Birkh¨auser, 1987 [Sh] Shubin, M. S.: Pseudodifferential Operators and Spectral Theory. Berlin-Heidelberg-New York:Springer-Verlag, 1987 [Sj] Si¨ostrand, J.: Semi-excited levels in non-degenerate potential wells. Asymptotic Analysis 6, 29–43 (1992) Communicated by B. Simon
Commun. Math. Phys. 257, 515–562 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1343-4
Communications in
Mathematical Physics
Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation ´ Christian Hainzl1,2 , Mathieu Lewin1 , Eric S´er´e1 1
CEREMADE, UMR CNRS 7534, Universit´e Paris IX Dauphine, Place du Mar´echal de Lattre de Tassigny, 75 775 Paris, Cedex 16, France. E-mail: {hainzl, lewin, sere}@ceremade.dauphine.fr 2 Laboratoire de Math´ematiques, Paris-Sud, Bˆat. 425, 91 405 Orsay Cedex, France Received: 29 March 2004 / Accepted: 13 January 2005 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: According to Dirac’s ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D 0 . In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane (J. Phys. B 22, 3791–3814 (1989)), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is the solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum. 1. Introduction The relativistic quantum theory of electrons and positrons is based on the free Dirac operator, which is defined by D 0 = −i
3
αk ∂k + β := −iα · ∇ + β,
k=1
where α = (α1 , α2 , α3 ) and
β= with σ1 =
01 , 10
I2 0 , 0 −I2
αk =
σ2 =
0 −i , i 0
0 σk , σk 0 σ3 =
1 0 . 0 −1
(1)
´ S´er´e C. Hainzl, M. Lewin, E.
516
We follow here mainly the notation of Thaller’s book [54]. We have chosen a system of units such that = c = 1, and also such that the mass me of the electron is normalized to 1. The operator D 0 acts on 4-spinors, i.e. functions ψ ∈ H := L2 (R3 , C4 ). It is selfadjoint on H, with domain H 1 (R3 , C4 ) and form domain H 1/2 (R3 , C4 ). Moreover, it is defined to ensure (D 0 )2 = − + 1. The spectrum of D 0 is (−∞; −1] ∪ [1; ∞). In what follows, the projector associated with the negative part of the spectrum of D 0 will be denoted by P 0 : P 0 := χ(−∞;0) (D 0 ). We then have
√ √ D 0 P 0 = P 0 D 0 = − 1 − P 0 = −P 0 1 − , √ √ D 0 (1 − P 0 ) = (1 − P 0 )D 0 = 1 − (1 − P 0 ) = (1 − P 0 ) 1 − ,
and 0 0 H = H− ⊕ H+ , 0 := P 0 H and H0 := (1 − P 0 )H. where H− + The fact that the spectrum of D 0 is not bounded from below is the source of many difficulties in Relativistic Quantum Mechanics. To explain why a free electron does not dissolve into the lower continuum, Dirac’s idea [13, 14] was to postulate that in the absence of external field, the vacuum contains infinitely many virtual electrons which completely fill up the negative part of the spectrum of D 0 . This Dirac Sea should be seen as an infinite Slater determinant 0 = ψ10 ∧ · · · ∧ ψi0 ∧ · · · , where (ψi0 )i≥1 is an 0 , whose density matrix is precisely orthonormal basis of H− |ψi0 ψi0 |. P0 = i≥1
The projector P 0 is often called the bare vacuum [9]. Let us now add an external coulomb field, created for instance by a system of smeared nuclei. The density of protons in this system is a nonnegative function1 n such that R3 n = Z, the total number of protons in the nuclei. In our system of units, the external 1 coulomb potential felt by the electrons is −αϕ, where ϕ = n ∗ |·| and α is a small dimensionless coupling constant, usually called the Sommerfeld fine structure constant. The Dirac operator with this external field is D αϕ := D 0 − αϕ.
(2)
Dirac postulated that the charge of the bare vacuum is not measurable. Indeed, P 0 commutes with translations, so its density of charge must be constant, and cannot create any electric force. However, in the presence of an external field, the virtual electrons react, by occupying the negative energy states of a new Dirac operator which does not 1 However, we shall not limit to nonnegative L1 densities n in this paper, since the model we want to study is able to describe the vacuum interacting with both matter and antimatter.
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
517
commute with translations: the vacuum is polarized. This polarization of the dressed vacuum, which takes the form of a local density of charge, is measurable in practice. The vacuum polarization plays a minor role in the calculation of the Lamb shift for the ordinary hydrogen atom (comparing to other electrodynamic phenomena), but it is important for high-Z atoms [43] and even plays a crucial role in muonic atoms [18, 24]. It also explains the production of electron-positron pairs, observed experimentally in heavy ions collisions [1, 46, 37, 50, 20]. When the external field is not too strong, a good approximation (called the Furry picture [21]) is to define the polarized vacuum as the projector P αϕ := χ(−∞;0) (D αϕ ). Note that in reality, the polarized vacuum modifies the electrostatic field, and the virtual electrons react to the corrected field. This remark naturally leads to a self-consistent equation for the dressed vacuum, and to a fixed-point iterative procedure for solving it. If one starts the procedure from P 0 , the first iteration gives P αϕ , and this explains why the Furry picture is a good approximation. But in practice, corrections to the Furry picture are necessary for high accuracy computations of electronic levels near heavy nuclei. These corrections can be interpreted as the second iteration in a Banach fixed-point algorithm (see, e.g., [43, Sect. 8.2]). In physics, self-consistent equations are usually derived as Euler-Lagrange equations of an energy functional. It is the case, for instance, in the nonrelativistic Hartree-Fock model [40]. Similarly, the self-consistent equation for the vacuum has a variational interpretation: it is satisfied by a minimizer of the Bogoliubov-Dirac-Fock (BDF) energy functional. This functional was first introduced by Chaix and Iracane [9], as a possible cure to the fundamental problems associated with standard relativistic quantum chemistry calculations. In these calculations, electrons near heavy nuclei are usually treated, in first approximation, by the Dirac-Fock model [53], a variant of Hartree-Fock in which the kinetic energy operator −/2 is replaced by the free Dirac operator D 0 . This approach gives results that are in excellent agreement with experimental data [34, 26, 12, 41]. When a higher accuracy is needed, the more sophisticated multiconfiguration Dirac-Fock-model is used to take into account correlation effects [25], and one can even compute the small corrections predicted by QED (vacuum polarization and radiative corrections), using perturbation methods. However, the Dirac-Fock model suffers from an important defect: the corresponding energy is not bounded from below, contrary to the nonrelativistic Hartree-Fock case, and this leads to important computational difficulties (see [9] for a discussion and detailed references). From the mathematical viewpoint, one can prove that the Dirac-Fock functional has critical points which are solutions of the Dirac-Fock equations [15, 44], but these critical points have an infinite Morse index, and the rigorous definition of a ground state is delicate [16, 17]. The second problem with Dirac-Fock is its physical derivation: one would like to interpret this model as a variational approximation of Quantum Electrodynamics (QED), which is believed to be the exact theory. An interesting attempt in this direction has been made by Mittleman [42], but it is not fully convincing. According to this author, the ground state of the Dirac-Fock model should be obtained by means of a max-min procedure applied to the no-photon QED Hamiltonian HQED . In this procedure, the reference projector for the normal ordering of HQED is not fixed, and the vacuum polarization terms are neglected. Then one has to look for a projector which maximizes, in the HartreeFock approximation, the ground-state energy of the normal-ordered Hamiltonian. From
´ S´er´e C. Hainzl, M. Lewin, E.
518
a mathematical viewpoint, Mittleman’s max-min principle has been investigated in the papers [4, 17, 7, 6]. In the case of zero or one electron [4, 17], it works very well and one shows that the projector P αϕ of the Furry picture is the optimal reference. But it seems, from the counterexample given in [6], that serious problems occur when there are several electrons. In their work [9], Chaix and Iracane derive their new mean-field model (which they call Bogoliubov-Dirac-Fock) from the no-photon QED Hamiltonian, normal-ordered with respect to a fixed reference: the free projector P 0 . They keep the vacuum polarization terms, pointing out that they are “necessary for the internal consistency of the relativistic mean-field theory and should therefore be taken into account in proper self-consistent calculations, independently of the magnitude of the physical effects” [9, p. 3813]. This allows them to obtain a bounded-below energy: the ground states are simply defined as minimizers and no max-min procedure is needed. A minimizer without charge constraint, if it exists, is a projector satisfying a self-consistent equation: it should be the negative spectral projector of the mean-field Hamiltonian generated by the nuclear charge density, corrected by a vacuum polarization effect. This self-consistent projector is the stable dressed vacuum. Now, if one restricts the BDF functional to the charge sector −N , and if one can find a minimizer, it will be solution of the Dirac-Fock equations for N electrons, corrected by a vacuum polarization term [9, Sect. 4.2]. The Dirac-Fock model is thus interpreted by Chaix-Iracane as a nonvariational approximation of BDF. In other words, the Euler-Lagrange equations only differ by small terms, but the variational structure is completely different since the DF functional is strongly indefinite (i.e., it is not bounded below and all its critical points have an infinite Morse index). As we have seen, the Chaix-Iracane model has several advantages as compared to the standard Dirac-Fock model: it is more accurate (taking into account vacuum polarization effects), its physical derivation is more convincing, and the ground state solutions have a simple definition as minimizers of the BDF functional. The drawback is that it is not easy to give a meaning to the quantities (energy of the vacuum, charge density of the vacuum) appearing in the BDF model. It is well known that there are divergent quantities in QED even after normal ordering, but Chaix and Iracane do not address this problem in their work. The first rigorous works on the BDF model are due to Chaix-Iracane-Lions [10] and Bach-Barbaroux-Helffer-Siedentop [4]. In particular, in [4], the authors give a rigorous meaning to the BDF energy in the class of operators with trace, and show that it is bounded below if one fixes the reference for normal ordering. Then Bach et al. vary the reference for normal ordering and neglect the vacuum polarization terms, which are experimentally small and mathematically divergent. This approximation is exactly the one made by Mittleman [42] in his formal derivation of the Dirac-Fock model. In the present work, our approach is different: we keep P 0 as reference for normal ordering, we study the full Chaix-Iracane model of the dressed vacuum (without neglecting any divergent term), and we control the divergences thanks to a momentum cut-off. Nevertheless, the paper [4] has been an important source of inspiration in our study: it contains very useful mathematical ideas and results, in particular the lower bound on the energy (see Theorem 1 in the present paper, which is a mere rephrasing, in our framework, of this estimate). Mathematically speaking, we shall say that a vacuum is an orthogonal projector P with the additional requirement that Q = P − P 0 ∈ S2 (H) , where S2 (H) denotes the class of Hilbert-Schmidt operators on H. As explained in the Appendix, this condition guarantees that P is the (unrenormalized) density matrix of
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
519
a dressed vacuum in the electron-positron Fock space associated with the free projector P 0 . This dressed vacuum may be seen (formally) as an infinite Slater determinant = ψ1 ∧ · · · ∧ ψi ∧ · · · , where (ψi )i≥1 is an orthonormal basis of P H. Since the model takes the free vacuum as reference according to Dirac’s ideas [13, 14], Q is the true (renormalized) one-body density matrix of . Following [9] (with notations from [4]), the BDF energy of the dressed vacuum can be written (formally) as follows, as a function of its renormalized density matrix: α ρQ (x)ρQ (y) E(Q) = tr(D 0 Q) − α ρQ ϕ + dx dy 2 |x − y| α |Q(x, y)|2 − dx dy (3) 2 |x − y| with ρQ (x) = tr C4 Q(x, x). By formal computations, Chaix and Iracane [9, Sect. 4.2] show that the Euler-Lagrange equation of this functional is [P , DQ ] = 0 ,
(4)
where DQ := D αϕ + αρQ ∗
1 Q(x, y) −α , |·| |x − y|
(5)
Q = P −P 0 . For a minimizer, the second order condition implies a more precise relation between P and DQ , which takes the form of a fixed-point equation: 1 Q(x, y) P = χ(−∞;0) (DQ ) = χ(−∞;0) D αϕ + αρQ ∗ −α . |·| |x − y|
(6)
Remark that if ϕ = 0 (no external potential), then P 0 is already a solution of this equation since Q = 0 in this particular case. The present paper contains the first mathematical study of a fixed-point algorithm for finding a solution of (6). Notice that the use of a fixed-point method to solve a selfconsistent equation is very common in quantum chemistry and physics and that most of the numerical algorithms used in practice are based on this idea. For a mathematical existence result using the Schauder fixed-point theorem, see the resolution of the Hartree equations in [57]. See also [8], where rigorous results are given on the convergence of standard Hartree-Fock iteration schemes. For the determination of a projector in no-photon QED, the fixed-point method has been used for the first time by Lieb and Siedentop [39]. Their goal was to replace P 0 by a new (self-consistent) projector commuting with translations, as reference for normal ordering in the absence of external field. We use the Banach fixed-point theorem as in [39], but our physical model is very different, and the necessary estimates are much more delicate in our case. Of course, we have to make an assumption on the external potential: it should have a certain regularity, and should not be too strong, otherwise we are not able to prove that the iteration method converges. If one is only interested in the existence of a minimizer, it is possible to remove the smallness assumption on the potential, but for this purpose the constructive fixed-point approach must be replaced by a direct – and non-constructive – minimization argument [28]. The regularity assumption cannot be dropped: this is a well known phenomenon in QED when P 0 is chosen as reference for normal ordering
´ S´er´e C. Hainzl, M. Lewin, E.
520
(see e.g., [35]). But this regularity is not really a restriction from the point of view of physics: point-like nuclei do not exist in nature. In [4], the operator D 0 Q is assumed to be trace class, so that the expressions (3) and (5) are well defined. Unfortunately, it turns out that, when ϕ is nonzero, Q = P − P 0 is never trace class if P is a solution of (6). Therefore no minimizer can exist in the trace class S1 (H) in the presence of an external field. So we must try to define the BDF energy and the self-consistent equation for operators which are not trace class, and this leads to several difficulties. A first problem occurs with the definition of tr(D 0 Q) in (3). To solve it, we will have to extend the trace functional to a bigger class of compact operators, namely the operators with “P 0 -trace” (see Sect. 2.1 below). A second problem occurs with the definition of the density ρQ . For this reason, we introduce a momentum cut-off , which means we replace the ambient space H by H := f ∈ H, supp(f) ⊂ B(0, ) . Since D 0 is a multiplication operator in Fourier space, H is invariant under P 0 and we keep the notation P 0 for the restricted operator. With the cut-off, the integral kernel Q(x, y) of P − P 0 ∈ S2 (H ) becomes smooth for any dressed vacuum P , and one can easily define ρQ (x) = tr C4 Q(x, x). Notice that, even with our ultraviolet cut-off, Q = P − P 0 is never trace class if P is a solution of (6) and if an external potential is present. As we √shall see it later on, our results will be valid under a technical condition of the form α ln ≤ C for some constant C. For a small α, this leads to an extremely large , which corresponds to scales that are far beyond the reach of experimental and theoretical physics at the present time. But our conditions do not allow to pass to the limit of an infinite cut-off. Note that if one expands the right-hand side of Eq. (6) in powers of the small parameter α, the first order term contains an expression which diverges logarithmically as goes to infinity. When the exchange term Q(x, y)/|x − y| is neglected, a simple algebraic manipulation allows to rewrite a posteriori our cut-off version of Eq. (6) in a renormalized form, with the divergent term removed, and the “bare” constant α in front of the charge densities replaced by a smaller, “dressed”, coupling constant αdr
α 1+
2α 3π
log
(details will be given in a forthcoming paper [28]). The dressed constant is the observable one. Its experimental value is αdr 1/137. This kind of “charge renormalization" associated with a momentum cut-off is standard in the physics literature (see, e.g., [33, Eq. (7.18)]). With this interpretation, the limit case of an infinite cut-off appears as unphysical (it would correspond to αdr = 0, which means no more electrostatic interaction). Remark 1. In the Furry picture, that is to say when P = P αϕ , it is known since the very beginning of QED [14, 31, 23, 55, 51] that the density ρ αϕ associated with Qαϕ = P αϕ −P 0 is never well-defined if no ultraviolet cut-off is imposed. One possible regularization procedure [19, 37, 29] is to remove the divergent part of ρ αϕ , which is (formally) αϕ proportional to the nuclear charge density n. This gives a renormalized density ρren which can be defined without the help of a high momentum cut-off. This procedure has recently been clarified by Hainzl and Siedentop in [29]. Some interesting features of
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
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αϕ
ρren , in the case of strong external fields, were obtained by Hainzl in [27]. We do not αϕ want to give a precise definition of ρren here and we refer the reader to [29, 27]. It would be tempting, instead of using a cut-off, to renormalize ρQ a priori in Eq. (6), as in [29]. But we do not know how to solve the resulting renormalized Eq. (6) if no cut-off is made. Moreover, even if we could find a solution without momentum cut-off, its interpretation as a minimizer of the BDF energy would be unclear. The paper is organized as follows. In the next section, we define the BogoliubovDirac-Fock model and state our main results. For the sake of clarity, we have brought all the proofs together in Sects. 3 and 4. In the Appendix, we explain in our language, for the reader’s convenience, how the BDF energy is deduced from no-photon QED by Chaix-Iracane in [9]. 2. Model and Main Results In this section, we study the Bogoliubov-Dirac-Fock model introduced in [9, 11]. Our system of notation is similar to [4], with the difference that we keep all the terms describing the vacuum polarization. This forces us to deal with operators which are not trace class, unlike [4]. 2.1. An extension of the trace functional. In order to give a meaning to the expression “tr(D 0 Q)” even when Q is not trace-class, we need the notion of “P 0 -trace”. In this section only, we work in an abstract Hilbert space h. Definition 1. Let P be a projector such that P and 1 − P have infinite rank, and A ∈ S2 (h). We shall say that A is P -trace class if and only if A++ := (1 − P )A(1 − P ) and A−− := P AP are trace class. Then we define the P -trace of A by tr P (A) := tr(A++ ) + tr(A−− ). We denote by SP1 (h) the space of all Hilbert-Schmidt operators which are P -trace class. Notice that if A is a trace class operator, then A ∈ SP1 (h) and tr(A) = tr P (A) for any projector P .
Remark 2. In [54, Sect. 5.7.2], a similar definition in connection with supersymmetry is made and the name “supertrace” is used. The following result, whose proof is given in Sect. 3, will be used repeatedly in the sequel. Lemma 1. Let P and P be two projectors such that P −P ∈ S2 (h). Then A is P -trace class if and only if it is P -trace class, and in this case tr P (A) = tr P (A). Another useful fact is that when A is Hilbert-Schmidt and A + P is a projector, then A has a P -trace, as explained below: Lemma 2. Let P and P be two projectors on a Hilbert space, such that P − P is a Hilbert-Schmidt operator. Then P − P is P -trace class. Moreover, tr P (P − P ) is an integer which satisfies
tr P (P − P ) = tr (P − P )2n+1 for all n ≥ 1, and tr P (P − P ) = 0 when P − P S < 1. ∞
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In our framework, a consequence is that, for any vacuum P such that Q = P − P 0 ∈ S2 (H ), tr P 0 (Q) is an integer which can be interpreted as the charge of the dressed vacuum P (see the Appendix for comments in this direction). When P solves the selfconsistent equation (6) and ϕ is not too strong, we will see that P is close to P 0 , so that its charge will be zero, according to the lemma. 2.2. The Bogoliubov-Dirac-Fock model. As in [4], we are going to extend the BDF energy to a convex set of compact operators, which can be interpreted as one-particle density matrices of quasi-free states. This kind of extension is standard for mean-field models depending only on the one-body density matrix (see [38, 3, 5]). In the whole paper, we assume that the nuclear charge density n = −ϕ/4π belongs to the Hilbert space
C = f ∈ L2 (R3 , R), D(f, f ) < ∞ , where f (k) g (k) D(f, g) = 4π dk. |k|2 We will choose the following Hilbert norm on C : ||f ||C :=
1 + |k|2 2 |f (k)| dk |k|2
1/2 .
The Bogoliubov-Dirac-Fock energy is defined by E( ) = tr P 0 (D 0 ) − αD(ρ , n) + on the set
α α D(ρ , ρ ) − 2 2
| (x, y)|2 dx dy |x − y|
0 G := ∈ SP1 (H ) | − P 0 ≤ ≤ 1 − P 0 , ρ ∈ C .
In (7), ρ (k) =
1 (2π)3/2
|p|≤
(7)
(8)
Tr C4
(p + k/2, p − k/2) dp
is the Fourier transform of the charge density ρ , which, formally, is the diagonal of
∈ S2 (H ), as explained in the Introduction. Thanks to the momentum cut-off, ρ
is compactly supported, so that ρ ∈ L1 , hence 1 ρ (x) = Tr C4 ( (x, x)) = Tr C4
(p, q) eix(p−q) dp dq . (9) (2π)3 |p|,|q|≤ Clearly, the function ∈ S2 (H ) → ρ ∈ C00 ∩ L2 (R3 ) is continuous. Notice that if for instance f, g ∈ H 1 (R3 , R), then the electrostatic energy is simply f (x)g(y) D(f, g) = dx dy, R3 ×R3 |x − y|
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but for functions in C, it does not necessarily have a meaning as a Lebesgue double integral in direct space. Note that the set G is convex, and that the elements of P 0 + G are not necessarily projectors. In fact, it is an easy exercise to show that an element of G is extremal if and only if it is of the form P − P 0 , with P a projector. It then follows from Lemma 2 that the set of all extremal points of G coincides with Q = P − P 0 , where
P = P orth. projector | Q = P − P 0 ∈ S2 (H ), ρQ ∈ C . It will turn out that, under some assumptions on α, and n, the BDF functional has a unique minimizer on G which is extremal. As a consequence, inf{E(P − P 0 ), P ∈ P } = inf{E( ), ∈ G } . In the next subsection, we give necessary and sufficient conditions satisfied by a minimizer of E. 2.3. Study of the BDF energy. We first state the following result, which is an easy translation, in our framework, of the stability estimate proved by Bach et al [4] (see also [10]): Theorem 1. Let be n ∈ C. Then 1. E is well-defined on G ; 2. if 0 ≤ α ≤ π4 , then ∀ ∈ G ,
E( ) +
α D(n, n) ≥ 0, 2
(10)
and therefore E is bounded from below on G , independently of ; 3. if 0 ≤ α ≤ π4 and n = 0, then E is non-negative on G [4, 10], 0 being the unique minimizer. Remark 3. Note that the result is optimal in the sense that the functional becomes unbounded from below when n = 0 if α > 4/π, as shown in [10 and 32]. Remark 4. Since αD(n, n)/2 in (10) is the electrostatic energy of the field created by n, (10) means that the total energy of the system is nonnegative. Proof of Theorem 1. We only explain here why E is well defined on G , the rest being identical to the proof of Theorem 1 in [4] (see [4, Eq. (18)-(19)]). 0 If ∈ SP1 (H ), then we have P 0 D 0 P 0 = D 0 P 0 P 0 = D 0 ++ ∈ S1 (H ) √ since P 0 commutes with D 0 and |D 0 | ≤ 1 + 2 . With a similar argument for 1 − P 0 , 0 we obtain that D 0 ∈ SP1 (H ). Therefore, tr P 0 (D 0 ) is well-defined and tr P 0 (D 0 ) = tr(D 0 ++ ) + tr(D 0 −− ) = tr(|D 0 | ++ ) − tr(|D 0 | −− )
(11)
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(notice that, due to the constraint −P 0 ≤ ≤ 1 − P 0 , one has ++ ≥ 0 and −− ≤ 0). On the other hand we have by Kato’s inequality π | (x, y)|2 dx dy ≤ tr(|D 0 | 2 ) |x − y| 2 showing that this last term is well-defined since |D 0 | is bounded on H and
∈ S2 (H ). We are interested in minimizers of the BDF functional, and we expect them to be in the class Q = P − P 0 . This leads to the following definition Definition 2. We say that a projector P is a BDF-stable vacuum if and only if P − P 0 is a minimizer of E on G . When there is no external potential, P 0 is the unique BDF-stable vacuum [10, 4], which corresponds to Dirac’s ideas. But if we consider a non-vanishing external potential 1 ϕ = n ∗ |·| , then P 0 obviously cannot be BDF-stable, since it is easy to create a state 0 −P ≤ γ ≤ 1 − P 0 such that E(γ ) < 0 = E(0). This means that the vacuum is necessarily polarized. More precisely, one can easily derive necessary conditions satisfied by a BDF-stable vacuum P . To this end, a perturbation of the form Q+γ = P −P 0 +γ , with γ ∈ S1 (H ) such that −P ≤ γ ≤ 1 − P is considered in Chaix-Iracane [9, formula (4.8)], and the energy E(Q + γ ) is expanded to get α α |γ (x, y)|2 dx dy + E(Q), (12) E(Q + γ ) = tr DQ γ + D(ργ , ργ ) − 2 2 |x − y| a formula which is valid when γ ∈ S1 (H ), the operator DQ being defined in (5). Remark 5. In [4, Formula (21)] and [7], the polarization potentials appearing in DQ and the energy of the vacuum E(Q) were neglected by the authors who used the following functional αϕ α α |γ (x, y)|2 ˜ EP (γ ) = tr D γ + D(ργ , ργ ) − dx dy, (13) 2 2 |x − y| with the constraints γ ∈ S1 and −P ≤ γ ≤ 1 − P (and even P γ (1 − P ) = 0 in [7]). Then a procedure taking the form supP inf −P ≤γ ≤1−P E˜P (γ ), related to Mittleman’s work [42], was considered in [4]. For the case of the vacuum (no constraint on the trace of γ ), the solution is the Furry picture P = P αϕ with γ = 0, as shown in [4]. We refer the reader to [9, p. 3809] and [17, 6, 7] for comments and results concerning Mittleman’s max-min in the case of N electrons (which corresponds to the additional constraint tr(γ ) = N ). From formula (12), it can be seen that a BDF-stable vacuum must satisfy the fixedpoint equation (6). The converse is also true under some assumptions: Theorem 2 (BDF-Stability). Let be P ∈ P and n ∈ C. We assume that there exists a positive constant d such that π d|DQ | ≥ |D 0 | with αd ≤ 1, (14) 4 where DQ is defined in (5). Then, the following assertions are equivalent
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1. P fulfills the equation P = χ(−∞;0) (DQ ) = χ(−∞;0) D
αϕ
1 Q(x, y) + αρQ ∗ −α . |·| |x − y|
(15)
2. P is the unique BDF-stable vacuum, i.e. P − P 0 is the unique global minimizer of E on G . The proof of this theorem will be given in Sect. 3. Some arguments are directly inspired of [4]. Remark 6. One can try to go further: if we translate the ideas of Chaix and Iracane in our language, the ground state of a molecule consisting of nuclei with total charge density n, surrounded by a cloud of N electrons, should solve the following constrained minimization problem min{E( ), ∈ G , tr P 0 ( ) = N }, with N ∈ N \ {0}. If this minimization problem has a solution Q, it will solve a selfconsistent equation of the form Q = χ(−∞;µ) (DQ ) − P 0 , where µ ∈ (−1; 1) is a Lagrange multiplier associated with the charge constraint, and interpreted as a chemical potential. For a not too strong external field ϕ, it should be possible to prove that µ ∈ (0; 1) and that the vacuum = χ(−∞;0) (DQ ) stays neutral, which means tr P 0 (−P 0 ) = 0. Therefore, we could split P = Q+P 0 = χ(−∞;µ) (DQ ) in the form P =+
N
|ψk ψk |.
k=1
The mono-electronic wave functions ψk would be solutions of the Dirac-Fock equations (with high momentum cut-off), perturbed by vacuum polarization terms: DQ ψk = εk ψk ,
0 < εk < 1 ,
1≤k≤N .
It is our goal to study this constrained variational problem in the near future. The present work, which deals with the unconstrained case, is a first step in this direction.
2.4. Existence of a BDF-stable vacuum. We may now state our main theorem. We recall that the norm on C is ||f ||C :=
1 + |k|2 2 |f (k)| dk |k|2
1/2 .
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Theorem 3 (Existence of a BDF-stable vacuum). Let be n ∈ C and b ∈ (0; 1). Then for all and α such that √ (16) 2 πα||n||C ≤ b and α ≤ αb (), where C(1 − b) αb () ∼→∞ √ , log there exists a unique BDF-stable vacuum P , which is a solution of 1 Q(x, y) 0 −α P = χ(−∞;0) DQ = χ(−∞;0) D + α(ρQ − n) ∗ |·| |x − y|
(17)
with Q = P − P 0 . Moreover, we have tr P 0 (Q) = 0. √ Remark 7. The first constraint 2 πα||n||C ≤ b means that the external field is not too strong. It explains why a neutral polarized vacuum is obtained (since tr P 0 (Q) = 0). In our proof, this constraint on the external field is necessary for the fixed point algorithm √ to converge. The second constraint α ≤ αb (), which essentially reduces to α log C(1 − b), is a technical condition due to our choice of norms, but we were |Q(x,y)|2 unable to drop it. It disappears if the exchange term α2 |x−y| dx dy is neglected in the energy, as can be seen from the proof. A precise definition of the constant C appearing in this result is given in the proof. Remark 8. There is an interesting symmetry property of the solutions of (17) when n is replaced by −n. Namely, if P is a solution of (17) with external density n, then P = Q + P 0 is a solution of (17) with external density −n, where Q = −CQC −1 , C being the charge conjugation operator [54, p. 14]. The two dressed vacua P and P have the same BDF energies and satisfy ρQ = −ρQ , as suggested by the intuition. For this symmetry between matter and antimatter to be true, it is essential to have the Fermi level at 0 and not at −1 (see, e.g., the comments of [49, p. 197] about this fact). 2.5. Idea of the proof of Theorem 3: the fixed-point algorithm. We end this section with a brief description of our fixed-point algorithm, used in the proof of Theorem 3, and which could be useful for practical computations. A natural scheme for solving (17) would be to construct a sequence (Qj )j ≥0 ⊂ G by taking Q0 = 0 and Qj (x, y) 1 Qj +1 = χ(−∞;0) D 0 + α(ρQj − n) ∗ −α (18) − P 0. |·| |x − y| Expanding this expression in powers of α, and considering the total density := ρQj − n, ρQ j
one can write the following recursion formula in Fourier space: n(k) − αB (k)ρ ρ Qj +1 (k) = − Qj (k) + α ρ1,0 (Qj )(k) +
∞
(Qj , ρQ ) (k) . (19) α n ρn j
n=2
The notations B , ρ1,0 and ρn are defined precisely in Sect. 4.1 (see Subsect. 4.1.3 and 4.1.4). The important point is that B (k) is a positive function which diverges logarithmically as → ∞ for any fixed k, whereas the other terms stay bounded. From
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
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(19) we thus see that the scheme (18) would converge under a condition of the form α log ≤ C. To improve this condition, we use a better algorithm in our proof. Our modified scheme consists in defining a sequence of pairs (Qj , ρj )j ≥0 ⊂ G × C such that (Q0 , ρ0 ) = (0, −n) and
Q
Qj (x, y) 1 ∗ −α j +1 = χ(−∞;0) D | · | − y| |x ρj +1 = L(α, )ρQ + 1 − L(α, ) ρ j j +1 0
+ αρj
− P0
(20)
(x) := ρ (x) − n(x) = tr where ρQ C4 Qj (x, x) − n(x), and L(α, ) is the linQj j ear operator which, in the Fourier domain, is just the multiplication by the function (1 + αB (k))−1 . The second equation in the iteration scheme (20) can be written in the form
−1 ρ − n(k) + α ρ 1,0 (Qj )(k) + j +1 (k) = (1 + αB (k))
∞
α n ρ n (Qj , ρj ) (k) . (21)
n=2
The divergent term now only appears in the denominator. So one expects a much better convergence. In the proof of Theorem 3, we show the convergence of the algorithm (20) under the conditions (16) but we believe that it converges independently of the cut-off . Qj (x,y) It can be seen from our proof that this holds when the exchange term α |x−y| is neglected in (20). In this case, the algorithm converges independently of to the solution of a reduced fixed-point problem (without exchange term), which is the unique minimizer on G of the convex functional Ered ( ) = tr P 0 (D 0 ) − αD(ρ , n) +
α D(ρ , ρ ) . 2
3. Proof of Theorem 2 In this section, we prove Theorem 2. To this end, we first need to prove Lemmas 1 and 2.
3.1. Proof of Lemmas 1 and 2. Proof of Lemma 1. Let P and P be two projectors such that P −P ∈ S2 (h), and a Hilbert-Schmidt operator A which is P -trace class. This means that P AP and (1 − P )A(1 − P ) are trace class. Let us first show that P AP is trace class. To this end, we write P AP = (P − P + P )A(P − P + P ) = (P − P )A(P − P ) + (P − P )AP + P A(P − P ) + P AP .
(22)
This shows that P AP is trace class since the last term is in S1 by assumption, P − P and A are in S2 , and P is bounded. The same computation shows that (1−P )A(1−P ) is trace class.
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We now compute tr[P AP ] = tr[(P − P )A(P − P )] + tr[(P − P )AP ] + tr[P A(P − P )] + tr[P AP ] = tr[A (P − P )(P − P ) + P (P − P ) + (P − P )P ] + tr[P AP ] = tr[A(P − P )] + tr[P AP ], where we have used the formula tr(AB) = tr(BA), valid for A, B ∈ S2 . The same computation gives tr[P+ AP+ ] = tr[A(P+ − P+ )] + tr[P+ AP+ ] = − tr[A(P − P )] + tr[P+ AP+ ], where we have used the notation P+ = 1 − P and P+ = 1 − P . Summing these two results, we obtain the formula tr P [A] = tr P [A]. Proof of Lemma 2. We introduce B = P − P . We have B 2 = P − P P − P P + P = (1 − P )B(1 − P ) − P BP . This implies that (1 − P )B(1 − P ) and −P BP are non-negative trace class operators. We now use the proof of [2, Theorem 4.1]. Since B ∈ S2 , we infer B 3 ∈ S1 and so (P , P ) is a Fredholm pair, in the language of [2]. Therefore, tr(B 3 ) is an integer and satisfies tr(B 3 ) = tr(B 2n+1 ) for all n ≥ 1. Now we have B 3 = B 2 P − B 2 P = P BP + P BP . Applying this result to 1 − P and 1 − P , we find B 3 = (1 − P )B(1 − P ) + (1 − P )B(1 − P ). Summing this two identities, we obtain by Lemma 1 2 tr(B 3 ) = tr P (B) + tr P (B) = 2 tr P (B). This shows that tr P (B) indeed equals the of index the pair of projectors (P , P ) defined in [2], an integer which vanishes when P − P S < 1 by the results of [2]. ∞
3.2. Preliminaries. To prove Theorem 2, we also need the following Lemma 3. Assume that ϕ = ρ ∗ ||∇ϕ||H 1 = 4π ||ρ||C ,
1 |·|
for some ρ ∈ C. Then
||ϕ||L∞ ≤ C∞ 4π ||ρ||C ,
||ϕ||L6 ≤ C6 4π ||ρ||C ,
where C∞ := 2π11/2 and C6 is the Sobolev constant for the inequality ||u||L6 (R3 ) ≤ C6 ||∇u||L2 (R3 ) . Proof. We have
1 + |k|2 1 2 | ρ (k)| dk = |k|2 (1 + |k|2 )| ϕ (k)|2 dk, 2 2 3 3 |k| (4π) R R
and so 1 1 ϕˆ 1 ≤ L 3/2 (2π ) (2π)3/2
dk 2 (1 + |k|2 ) 3 |k| R The rest is easily obtained by the Sobolev inequalities. ||ϕ||L∞ ≤
1/2 4π ||ρ||C .
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
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Lemma 4. Let P be a projector in P and Q = P − P 0 . Then DQ is bounded. Proof. Due to the cut-off in Fourier space, D 0 is bounded on H . On the other hand, if 1 ϕ = ρ ∗ |·| for some ρ ∈ C, then ϕ ∈ L∞ by Lemma 3 and so this is also a bounded
operator. Let us now denote R(x, y) = Q(x,y) |x−y| . We then have 2 Q(x, y)f (y) |Q(x, y)|2 |f (y)|2 dy ≤ dy × dy |Rf (x)|2 = |x − y| R3 R3 |x − y| R3 |x − y| and since, by Kato’s inequality,
|f (y)|2 π dy ≤ f, |D 0 |f 3 |x − y| 2 R 2 |Q(x, y)| π dx dy ≤ tr(|D 0 | Q2 ) 6 |x − y| 2 R
this shows that R ≤ C |D 0 |1/2 and so R is bounded.
(23)
Lemma 5. Let P be a projector in P and Q = P − P 0 . Then DQ ∈ SP1 0 (H ) for all ∈ G and we have Q(x, y) 0
= tr P 0 (DQ ). tr P 0 (D ) + αD(ρQ − n, ρ ) − α tr |x − y| Q(x,y) Q(x,y)
(x,y) |x−y| is trace class, since |x−y|1/2 and |x−y|1/2 are in S2 by 1 1 and P = χ(−∞;0) (D). = D −α(ρQ −n)∗ |·| = D 0 +α(ρQ −n)∗ |·|
Proof. Remark that R =
(23). Let us now define D By the result of Klaus-Scharf [36] (see also [30] and the proof of Theorem 3), it is known that P − P 0 ∈ S2 (H ). Thus P D P = DP P ∈ S1 (H ) since ∈ SP1 0 = S1P by Lemma 1, and D is bounded by the proof of Lemma 4. Therefore, DQ = D + R is in SP1 0 . To show the expected equality, we prove tr P 0 (D 0 ) + αD(ρQ , ρ ) = tr P 0 (D ),
(24)
= ρ − n ∈ C. This will end the proof since the other term is trace class. The where ρQ Q general idea of the proof is to approximate by a trace class operator for which this equality is true, and to pass to the limit. However, the behaviour of the associated density in the space C is not obvious and to overcome this difficulty, we shall also approximate to obtain a potential in L2 (R3 ). We thus start by choosing a sequence ρ the density ρQ j in C, such that ϕ = ρ ∗ 1 is in L2 (R3 ). We can which converges as j → +∞ to ρQ j j |·| (k)χ choose for instance ρj (k) = ρ . We now show Q
(|k|≥1/j )
tr P 0 (D 0 ) + α
R3
ρ ϕj = tr P 0 (Dj )
(25)
n for all ∈ G , and where Dj = D 0 − αϕj . To this end, we may find a sequence +− 0 0 of finite rank operator which converges to +− = (1 − P ) P in S2 . Then n
++ +−
n := n )∗ ( +− −−
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530
converges to in S2 . Since n ∈ S1 for all n ≥ 0, we have 0 n ρ n ϕj = tr P 0 (Dj n ). tr P 0 (D ) + α
(26)
R3
By (9), the function Q ∈ S2 (H ) → ρQ ∈ L2 (R3 ) is continuous. Therefore, ρ n → ρ in L2 (R3 ). Since ϕj ∈ L2 (R3 ), we may now pass to the limit in (26) and obtain 0 n 0 n ρ ϕj = tr P 0 (D ) + α ρ ϕj , lim tr P 0 (D ) + α n→∞
R3
R3
where we have used that n n ) + tr(D 0 −− ) = tr(D 0 ++ ) + tr(D 0 −− ) = tr P 0 (D 0 ). tr P 0 (D 0 n ) = tr(D 0 ++
Let us now pass to the limit in the right-hand side. Indeed, we can write, by Lemma 1, tr P 0 (Dj n ) = tr Pj (Dj n ) = tr(Dj Pj n Pj ) + tr(Dj (1 − Pj ) n (1 − Pj )), where Pj = χ(−∞;0) (Dj ) and since Pj − P 0 ∈ S2 by [36]. Now, using (22), it is easily seen that Pj n Pj → Pj Pj and (1 − Pj ) n (1 − Pj ) → (1 − Pj ) (1 − Pj ) in S1 as n → ∞, since these terms can be expanded as a sum of trace class operators and products of at least two Hilbert-Schmidt operators converging strongly in S2 . Since Dj is bounded by the proof of Lemma 4, we obtain that tr Pj (Dj n ) →n→∞ tr Pj (Dj ) = tr P 0 (Dj ) by Lemma 1. As a conclusion, we have proved (25) for all ∈ G . To finish the proof, it remains to pass to the limit as j → +∞. Since ρ ϕj = D(ρ , ρj ) R3
strongly in C as j → ∞, we may pass to and ρ ∈ C (recall that ∈ G ), ρj → ρQ the limit in the left-hand side of (25). To pass to the limit in the right-hand side, we use again the fact that
tr P 0 (Dj ) = tr Pj (Dj ) = tr(Dj Pj Pj ) + tr(Dj (1 − Pj ) (1 − Pj )). By the results of Klaus-Scharf [36] (see also the proof of Theorem 3), it is known that in C. Using again (22), it is then easily seen that Pj − P → 0 in S2 , since ρj → ρQ Pj Pj → P P and (1 − Pj ) (1 − Pj ) → (1 − P ) (1 − P ) in S1 as j → ∞. Since Dj → D in S∞ by Lemma 4, we may thus pass to the limit and obtain the desired equality (24). 3.3. End of the proof of Theorem 2. We start by proving 1) ⇒ 2). We thus consider a projector P that satisfies the assumption of the theorem, and is also a solution to the equation P = χ(−∞;0) (DQ ). We fix some ∈ G and show that E( ) ≥ E(Q). To this end, we write E( ) = E(Q + ), where = − Q = + P 0 − P . By assumption,
fulfills −P 0 ≤ ≤ 1 − P 0 , and so fulfills −P ≤ ≤ 1 − P . Using Lemma 5, we may expand E(Q + ) and obtain
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
E(Q + ) = tr P 0
α α DQ + D(ρ , ρ ) − 2 2
531
| (x, y)|2 dx dy + E(Q). |x − y|
Using now Lemma 1, we see that it is thus sufficient to prove that α | (x, y)|2 α dx dy > 0, tr P DQ + D(ρ , ρ ) − 2 2 |x − y| for any ∈ SP1 (H ) such that = 0, ρ ∈ C and −P ≤ ≤ 1 − P , which is an easy adaptation of the proof of [4, Theorem 2]. We now show 2) ⇒ 1). Let P satisfy the assumption of the theorem, and such that Q = P − P 0 is a minimizer of E in B . We therefore have, by formula (12), α α |γ (x, y)|2 dx dy ≥ 0 (27) tr DQ γ + D(ργ , ργ ) − 2 2 |x − y| for all γ ∈ S1 (H ) such that −P ≤ γ ≤ 1 − P . The proof of Theorem 4 by Bach et al. [4] now implies that P = χ(−∞;0) (DQ ). Their proof is done with D αϕ instead of DQ but they also mention that it can be extended to a more general case, provided 0∈ / σ (DQ ) and P , 1 − P leave the domain of DQ invariant, which is the case here. 4. Proof of Theorem 3 In this section, we prove Theorem 3 by using a Banach fixed-point method. 4.1. Preliminaries. We start by defining the norms and spaces that will be used to apply this well known result. In fact, one of the main difficulties we faced in this work consisted in finding suitable Banach spaces. 4.1.1. Norms and spaces. We choose the following norms: 1/2 2 2 ||Q||Q := E(p − q) E(p + q)|Q(p, q)| dp dq ,
E(p − q)2 |R(p, q)|2 dp dq E(p + q) 1/2 E(k)2 2 | ρ (k)| dk , ||ρ||C := |k|2 1/2 2 2 2 |k| E(k) | ϕ (k)| dk , ||ϕ||Y :=
||R||R :=
where E(k) =
1/2 ,
1 + |k|2
and denote by Q, R, C and Y the associated Hilbert spaces. The dual space C of C will be also useful and we introduce 1/2 |k|2 ˆ (k)|2 dk ||ζ ||C := | ζ . E(k)2
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In the following, it will be easier to use the norm ||R||R , where R = Q(x,y) |x−y| in our estimates and a relation with ||Q||Q will then be needed. To this end, we first need the following well known lemma, which will be useful throughout the rest of the proof. Lemma 6. For all ξ and η in R3 , we have
∀s ≥ 0, E(ξ )s ≤ 2δ(s) E(ξ − η)s + E(η)s , |s|
(28)
|s|
∀s ∈ R, E(ξ ) ≤ 2 E(ξ − η) E(η) , s
with
δ(s) =
s
(29)
s if 0 ≤ s < 1 . s − 1 if s ≥ 1
Remark 9. A trivial consequence of (28) is the following inequality 1 1 1 ≤ min , . E(p) + E(q) E(p + q) E(p − q)
(30)
We shall also need the following Lemma 7. We have sup p,q∈R3
E(p + q) ≤ 2. E(p)2 E(p − q)2
(31)
Proof. Let us introduce the function f (p, q) = E(p + q)E(p)−2 E(p − q)−2 for (p, q) ∈ R3 × R3 . We have 0 ≤ f (p, q) =
2E(2p)E(p − q) 4 E(2p − (p − q)) ≤ ≤ E(p)2 E(p − q)2 E(p)2 E(p − q)2 E(p)E(p − q)
by Lemma 6. Therefore lim(p,q)→∞ f (p, q) = 0 and f attains its maximum on R3 ×R3 . Computing ∇q f (p, q), we see that at a critical point of f , p and q are always parallel. It therefore suffices to study the function g(x, y) = E(x + y)E(x)−2 E(x − y)−2 for (x, y) ∈ R × R. It is then easy to see that maxR2 g < 2 (the bound (31) is indeed not optimal). Now we can give a connection between ||R||R and ||Q||Q when R = recall the easy relation between ||ρ||C and ||ϕ||Y when ϕ = ρ ∗
1 |·| ).
Q(x,y) |x−y|
1 Lemma 8. If ρ ∈ C and Q ∈ Q, then we have ϕ = ρ∗ |·| ∈ Y and R(x, y) = and more precisely
||ϕ||Y = 4π||ρ||C , ||R||R ≤ CR ||Q||Q , √ R |D0 |−1 ≤ 2||R||R , S2
with CR :=
du 1 θ E(2x) . inf sup 1+θ |u − x|2 2π 2 θ∈(0;2) x∈R3 R3 E(2u)
(we also
Q(x,y) |x−y|
∈R
(32) (33)
(34)
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
533
Proof. We have 1 Q(p − l, q − l) R(p, q) = dl 2π 2 R3 |l|2 so we obtain, for some fixed θ ∈ (0; 2) E(p − q)2 ||R||2R = |R(p, q)|2 dp dq E(p + q) E(2v)2 =8 |R(u + v, u − v)|2 du dv E(2u) + v, l − v) · Q(l + v, l − v) 8 E(2v)2 Q(l = du dv dl dl 2 2 2 (2π ) E(2u) |l − u| |l − u|2 E(2v)2 8 du dv = (2π 2 )2 E(2u) 1+θ + v, l − v) E(2l ) 1+θ + v, l − v) 2 Q(l E(2l) 2 Q(l × dl dl 1+θ 1+θ 2 2 E(2l ) |l − u| |l − u| E(2l) |l − u| |l − u| + v, l − v)|2 E(2v)2 E(2l)1+θ |Q(l 8 ≤ du dv dl dl (2π 2 )2 E(2u) E(2l )1+θ |l − u|2 |l − u|2 + v, l − v)|2 Kθ (l) dv dl, ≤8 E(2v)2 E(2l)|Q(l where E(2l)θ Kθ (l) := (2π 2 )2
Now, let us introduce
Cθ := sup
x∈R3
Remark that
1 du dl . E(2u)E(2l )1+θ |l − u|2 |l − u|2 E(2x)
θ
du 1+θ |u − x|2 3 E(2u) R
.
du du 1 ≤ 1+θ θ , 1+θ |u − x|2 1+θ |u − e |2 3 3 E(2u) 2 |x| |u| x R R
where ex := x/|x|, showing that Cθ < ∞ when θ ∈ (0; 2). Now we have 1 1 E(2l)θ θ Kθ (l) = E(2u) du dl (2π 2 )2 E(2u)1+θ |l − u|2 E(2l )1+θ |l − u|2 1 Cθ 2 E(2l)θ du × Cθ ≤ , ≤ (2π 2 )2 E(2u)1+θ |l − u|2 2π 2 and so ||R||2R ≤ 8
Cθ 2π 2
2
+ v, l − v)|2 dv dl ≤ E(2v)2 E(2l)|Q(l
which ends the proof of (32).
Cθ 2π 2
2 ||Q||2Q
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To prove (33), we remark that we have, by (31), 2 R|D0 |−1
q)|2 |R(p, dp dq S2 E(p)2 q)|2 E(p − q)2 |R(p, ≤2 dp dq = 2||R||2R . E(p + q) =
4.1.2. An estimate from below for DQ . We now state a lemma in which we give a lower estimate for the operator DQ,µ := D 0 + αµ ∗
1 Q(x, y) −α |r| |x − y|
by D 0 , in terms of the spaces introduced above. For our result, we are interested in DQ = DQ,ρQ −n but this definition with an arbitrary density µ will be useful later on. Lemma 9. Assume that (Q, µ) ∈ Q × C are such that √
√ α 2 π||µ||C + 2CR ||Q||Q < 1.
(35)
Then DQ,µ is a bounded operator which satisfies
√ √ |DQ,µ | ≥ 1 − α 2 π||µ||C + 2CR ||Q||Q |D 0 |. Proof. We have, with ϕ = µ ∗
(36)
1 |·| ,
ϕ u 2 ≤ ϕ ∞ ||u||L2 ≤ 2π 1/2 ||µ||C |||D0 | · u||L2 L L by Lemma 3, and ||Ru||L2 = R|D0 |−1 |D0 |u
L2
≤ R|D0 |−1
S2
|||D0 |u||L2 ≤
√ 2CR ||Q||Q |||D0 | · u||L2
√ by (33). This shows that |ϕ − R| ≤ (2π 1/2 ||µ||C + 2CR ||Q||Q )|D 0 |, the square root being monotone. This proves that DQ,µ is bounded since D0 is bounded on H , and gives the expected inequality. Remark that Lemma 9 will be useful when we apply Theorem 2 (see the condition (14) in the statement). It also implies 0 ∈ / σ (DQ,µ ), a fact that will be used to compute the projection χ(−∞;0) (DQ,µ ).
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
535
4.1.3. Expansion by Cauchy’s formula. We want to solve the equation 1 Q(x, y) 0 −α − χ(−∞;0) (D 0 ) := F1 (Q). Q = χ(−∞;0) D + α(ρQ − n) ∗ |·| |x − y|
√ √ / σ (DQ ) by Lemma 9. We may If α 2 π ||ρQ − n||C + 2CR ||Q||Q < 1, then 0 ∈ thus use the method of [29] and expand F1 by Cauchy’s formula +∞ ∞ 1 1 1 F1 (Q) = − dη α n Qn , − = 2π −∞ DQ + iη D 0 + iη n=1
where
n 1 1 (RQ − ϕQ ) 0 dη 0 , D + iη D + iη −∞ 1 Q(x, y) = ρQ − n, ϕQ = ρQ ∗ , RQ (x, y) = . ρQ |r| |x − y|
1 Qn = − 2π
We shall write Qn =
+∞
Qk,l ,
k,l/ k+l=n
Qk,l
(−1)l+1 = 2π
+∞
I ∪J ={1,...,n}, −∞ |I |=k, |J |=l
n 1 1 Rj 0 , dη 0 D + iη D + iη j =1
if j ∈ J (Q where Rj = RQ if j ∈ I and Rj = ϕQ k,l is the sum of all the terms containing k RQ ’s and l ϕQ ’s). We also denote ρk,l := ρQk,l . Hence our equation can be written ∞ α n Qn (Q, ρQ ), Q = n=1 (37) ∞ n ρQ = α ρ (Q, ρ ), n Q n=1
= ρ − n. In order to have a where we recall that Qn and ρn depend on both Q and ρQ Q better condition on α and , we shall now change the second equation for the density, by taking into account the special form of the first order term ρ1 . To this end, we need to compute this term explicitly.
4.1.4. The first order density. Recall that +∞ 1 1 1 Q0,1 = ϕQ dη 0 0 2π −∞ D + iη D + iη so that −5/2 Q 0,1 (p, q) = (2π)
+∞ −∞
dη
1 1 ϕ . Q (p − q) α · p + β + iη α · q + β + iη
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536
We now introduce
1 +∞ 1 1 · dη π −∞ α · p + β + iη α · q + β + iη +∞ 1 α · p + β − iη α · q + β − iη = dη 2 · 2 π −∞ p + 1 + η2 q + 1 + η2 (α · p + β) (α · q + β) 1 −1 . = E(p) + E(q) E(p) E(q)
M(p, q) :=
(38)
Hence Q 0,1 (p, q) =
1 (p ϕ 5/2 2 π 3/2 Q
− q)M(p, q).
This enables us to compute
1 Tr C4 Q 0,1 (l + k/2, l − k/2) dl 3/2 (2π) |l|≤ 1 ϕ (k) = Tr C4 (M(l + k/2, l − k/2))dl 16π 3 Q |l|≤ 1 = − ϕ (k)|k|2 B (k) 4π Q = −ρ Q (k)B (k),
ρ 0,1 (k) =
where 1 B (k) = − 2 2 π |k|
|l|≤
(39)
(l + k/2) · (l − k/2) + 1 − E(l + k/2)E(l − k/2) dl. (40) E(l + k/2)E(l − k/2)(E(l + k/2) + E(l − k/2))
This function is computed in [45] B (k) =
1 π
E()
0
z2 − z4 /3 dz , 2 2 1 − z 1 + |k| (1 − z2 )/4
and it is logarithmically divergent since B (0) =
1 π
0
E()
z2 − z4 /3 2 5 2 dz = log() − + log 2 + O(1/2 ). 1 − z2 3π 9π 3π
4.1.5. Equation. We are now able to introduce the function on which we shall apply the fixed-point theorem. According to what we said above, the equation in ρQ can be written ρQ =
∞
α n ρn (Q, ρQ )
(41)
n=1 for simplicity) or equivalently (we forget the dependence in Q and ρQ ρ Q (k) = −αB (k)ρQ (k) + α ρ 1,0 (k) +
∞ n=2
α n ρn (k)
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
and ρ Q (k) = −
537
∞ 1 1 n α ρn (k) , n(k) + α ρ 1,0 (k) + 1 + αB (k) 1 + αB (k)
(42)
n=2
which is more adapted to a fixed-point argument, since the divergent term B (k) appears now in the denominator. = ρ − n and not for Notice that we now study an equation for the full density ρQ Q ρQ as previously. We therefore introduce the following space: X =Q×C consisting of all the pairs (Q, ρ ) such that Q ∈ Q and ρ ∈ C. Notice that in this space, ρ can be different from ρQ − n. However, we shall find a solution of the equations in this space, which satisfies ρ = ρQ − n. We also introduce on X the norm √ √ (Q, ρ ) = CR 2||Q||Q + 2 π ||ρ ||C , where we recall that CR is defined in Lemma 8. In the following, we shall keep the and not ρ . notation ρ to remind the reader that the equation indeed concerns ρQ Q We now introduce the function F : X → X defined by F (Q, ρ ) = FQ (Q, ρ ) , Fρ (Q, ρ ) , where FQ (Q, ρ ) = χ(−∞;0) (DQ,ρ ) − P 0 =
∞
α n Qn (Q, ρ ),
(43)
n=1
Fρ (Q, ρ ) = −
1 1 n+ 1 + αB 1 + αB
α ρ 1,0 (Q, ρ ) +
∞
α ρn (Q, ρ ) , (44) n
n=2
by ρ ). Remark that ρ = ρ Qn (Q, ρ ) and ρn (Q, ρ ) being defined in (37) (replace ρQ n Qn for all n ≥ 2. In the proof of Theorem 3, we solve the fixed-point equation in X ,
F (Q, ρ ) = (Q, ρ ). 4.2. Existence of a fixed-point of F . To prove our main theorem, we need the following estimates: Proposition 10. Assume that (Q, ρ ) ∈ X is such that 0 ∈ / σ (DQ,ρ ). Then we have +∞ n √ F (Q, ρ ) ≤ 2 π||n||C + κ1 ()α (Q, ρ ) + κn α (Q, ρ ) ,
(45)
n=2 +∞ n−1 F (Q, ρ ) ≤ κ1 ()α + α nκn α (Q, ρ ) , n=2
(46)
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538
where
√ √ √ √ log CR 2 CR 2 κ1 () = max log , 2CR + 3/2 √ log , ∼→∞ √ √ 2 π π π and (κn )n≥2 √ is a sequence of positive numbers independent of and which satisfies κn ∼n→∞ K n for some constant K. To prove this proposition, we have to do some tedious estimates. Before starting this proof, let us show that Theorem 3 follows from Proposition 10. n Proof of Theorem 3. We introduce the function f (x) = ∞ n=2 κn x , which is a power series with a radius of convergence equal to 1. The estimates (45) and (46) can be written √ F (Q, ρ ) ≤ 2 π||n||C + κ1 ()α (Q, ρ ) + f α (Q, ρ ) , (47) F (Q, ρ ) ≤ κ1 ()α + αf α (Q, ρ ) . To apply the Banach fixed-point theorem, we now have to find a ball B(0, R) ⊂ X which is invariant under the function F and on which F is a contraction. Let R > 0 be some fixed radius. We have F (Q, ρ ) ≤ κ1 ()α + αf (αR) := µ. sup (Q,ρ )∈B(0,R)
Moreover, we also have F (Q, ρ ) ≤ F (Q, ρ ) − F (0, 0) + ||F (0, 0)|| ≤ µ (Q, ρ ) + ||F (0, 0)|| . Therefore a condition for the ball B(0, R) to be invariant under the action of F is ||F (0, 0)|| ≤ (1 − µ)R. Notice that since F (0, 0) = (0, 0), this inequality also contains the contraction condition µ < 1. Additionally due to Lemma 9 we assume αR < 1 as well as απ ≤ 1, 4(1 − αR) due to Theorem 2 and Lemma 9 which is equivalent to α≤
1 . π/4 + R
As a conclusion, if (α, R) fulfills √ 2 π||n||C + αRκ1 () + αRf (αR) ≤ R 1 , α ≤ π/4 + R
(48)
then we are able to apply the Banach Fixed-Point Theorem on B(0, R). Remark that these inequalities also contain the conditions µ < 1 and αR < 1. Notice also that if (α, R) is a solution to (48), then (α , R) is a solution to (48) for all α ≤ α, since the function which appears on the left of (48) is increasing in α.
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
539
√ Now, if we assume that 2 πα||n||C ≤ b, we obtain that if (α, R) fulfills b + αRκ1 () + αRf (αR) ≤ R α 1 α ≤ , π/4 + R
(49)
then it also fulfills (48). The first inequality of (49) is simpler when it is written in terms of the variables α and x := αR. It becomes b x + κ1 ()x + xf (x) ≤ α α
(50)
which implies x ∈ [b; 1). Now, given b, and x let us call ab, (x) the maximal value of α such that (50) holds, i.e. ab, (x) =
x−b , κ1 ()x + xf (x)
(51)
which is defined for x in [b; 1). Since limx→1 ab, (x) = ab, (b) = 0, we may denote by x ∈ (b; 1) the largest maximizer of the function ab, in the interval [b; 1). We now define Rb () := x /ab, (x ) and 1 αb () := min ab, (x ), . π/4 + Rb () As a conclusion, for all 0 ≤ α ≤ αb (), (α, Rb ()) is a solution of (48). This means that F is a contraction on B(0, Rb ()), on which we can apply the Banach Theorem. This gives a unique solution to the equation F (Q, ρ ) = (Q, ρ ) in B(0, Rb ()) ⊂ X . Let us now show that P is indeed a solution to (6). In fact ρ is a solution to (42) and so ρ = ρ + n is a solution to (41). On the other hand, we have Q = χ(−∞;0) (DQ,ρ ) − P 0 , and (41) means exactly that ρ = ρQ . Hence, P is a solution to P = χ(−∞;0) (DQ ). Thanks to the proof, we know that P satisfies the assumptions of Theorem 2, and so P is the unique BDF-stable vacuum (i.e. P − P 0 is the unique global minimizer of the BDF energy). To end the proof, let us study the behaviour of αb () as → ∞. Computing dab, (x)/dx, we find that x must satisfy the equation κ1 () + f (x ) =
x (x − b) f (x ). b
Since κ1 () diverges as → ∞, we see that f (x ) → ∞ and therefore x → 1 as → ∞. Now, since f (x) = ox→1 f (x) , we obtain that f (x ) = o→∞ (κ1 ()). Thus
αb, (x ) ∼→∞ As a conclusion, αb () ∼→∞ with C =
√ π √ . 2CR
1−b κ1 ()
and
Rb () ∼→∞
κ1 () . 1−b
√ C(1 − b) 1−b π(1 − b) = √ ∼→∞ √ √ κ1 () log CR 2 log
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4.3. Proof of Proposition 10: Estimates. In this section, we prove the claimed estimates of Proposition 10. We will have to introduce many constants. For the sake of clarity, a guide is provided to the reader at the very end of the proof, Sect. 4.3.5. Remark first that we have 1 ≤1 1 + αB (k) for all k ∈ R3 . Therefore, to estimate the norm ||Fρ (Q, ρ )||C , it suffices to estimate the norms of ρ1,0 (Q, ρ ) and ρn (Q, ρ ), due to (44). 1 For (Q, ρ ) ∈ X , we introduce the notation R = Q(x,y) |x−y| ∈ R, and ϕ = ρ ∗ |·| . We then remark that n √ F (Q, ρ ) ≤ 2 π||n||C + α (Q1 , ρ1,0 ) + α ||(Qn , ρn )|| n≥2
and estimate each term separately. A similar argument can be done for F (Q, ρ ). 4.3.1. First order terms. Lemma 11. We have the following estimates: (log )1/2 ||ϕ ||Y = 2(log )1/2 ||ρ ||C , 2π CR (log )1/2 ||Q1,0 ||Q ≤ ||RQ ||R ≤ CR ||Q||Q , ||ρ1,0 ||C ≤ ||Q||Q . 4π Therefore (Q1 , ρ1,0 ) ≤ κ1 () (Q, ρ ) , ||Q0,1 ||Q ≤
where
√ √ √ log CR 2 κ1 () = max log , 2CR + 3/2 √ . √ π 2 π
(52)
Proof. Recall that 1 ϕ (p − q)M(p, q), (53) 25/2 π 3/2 where the matrix M(p, q) is defined in (38), and whose properties are summarized in the following Q 0,1 (p, q) =
Lemma 12. Let + (p) = α·p+β+E(p) and − (p) = −(α·p+β)+E(p) be the projections 2E(p) 2E(p) 4 0 matrices in C onto the eigenspaces of D in Fourier space. We then have 1 Tr C4 + (p)− (q) , E(p) + E(q) |M(p, q)|2 = Tr C4 M(p, q)M(p, q)∗ 1 =8 Tr 4 + (p)− (q) , (E(p) + E(q))2 C p·q +1 Tr C4 + (p)− (q) = Tr C4 − (p)+ (q) = 1 − . E(p)E(q) Tr C4 (M(p, q)) = −4
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
541
Moreover, we have |p − q|2 ∀p, q ∈ R , Tr C4 (p) (q) ≤ min ,2 . (54) 2E((p + q)/2)2 Proof of Lemma 12. We only prove (54). We have Tr C4 + (p)− (q) ≤
3
+
−
|+ (p)| |− (q)| = 2, so when t := Now we have 1−
|p−q|2 4E((p+q)/2)2
≥ 1, there is nothing to prove.
l2 − k2 + 1 p·q +1 p+q p−q = 1− with l = , k= E(p)E(q) E(l + k)E(l − k) 2 2 1−t = 1− , (1 + t)2 − 4zt 2
where t = |k|2 /E(l)2 and z = |k|(l·k) 2 (1+l 2 ) ∈ [0; 1). When t ∈ [0; 1) and z ∈ [0; 1), the expression above is decreasing in z and so we obtain 1−
p·q +1 2t 1−t = ≤1− ≤ 2t E(p)E(q) 1+t (1 + t)2
which ends the proof.
• Let us now treat Q0,1 . From (53), we obtain 2 |Q 0,1 (p, q)| =
1 25 π 3
|ϕ |2 (p − q)|M(p, q)|2 ,
and so
2 E(p − q)2 E(p + q)|Q 0,1 (p, q)| dp dq 1 = 5 3 dk duE(k)2 E(2u)|ϕ |2 (k)χ (|u| ≤ )|M(u + k/2, u − k/2)|2 2 π 1 4 2 2 2 dk|k| E(k) |ϕ | (k) ≤ 5 3 2 2 π |u|≤ E(2u)E(u)
by Lemma 12. Now we have √ du 1 3 1 = 4π ≤ 2π log argsh(2) + √ argth √ 2 2 3 |u|≤ E(2u)E(u) 1+4 (55) for ≥ 3. So we obtain ||Q0,1 ||Q ≤
(log )1/2 ||ϕ ||Y = 2(log )1/2 ||ρ ||C . 2π
• ρ1,0 and Q1,0 . We have Q1,0 = −
1 2π
+∞
−∞
dη
D0
1 1 R 0 + iη D + iη
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542
so that
+∞
1 1 q) dη R(p, α · p + β + iη α · q + β + iη −∞ 1 (α · p + β) (α · q + β) 1 R(p, q) − R(p, q) =− 2 E(p) + E(q) E(p) E(q)
−1 Q 1,0 (p, q) = −(2π)
and 2 |Q 1,0 (p, q)| ≤
1 1 q)|2 ≤ q)|2 , |R(p, |R(p, (E(p) + E(q))2 E(p + q)2
showing that ||Q1,0 ||Q ≤ ||R||R ≤ CR ||Q||Q . Now, we have 1 ρ Tr 4 Q1,0 l + 1,0 (k) = (2π )3/2 R3 C 1 l+ = − 5/2 3/2 Tr C4 R 2 π R3
k ,l − 2 k ,l − 2
k dl 2 k k k M l + ,l − χ (|l| ≤ )dl, 2 2 2
so we obtain |ρ 1,0 (k)| ≤
1/2 1 −1 2 E(2l) | R(l + k/2, l − k/2)| dl 25/2 π 3/2 R3 1/2 2 × E(2l)|M(l + k/2, l − k/2)| dl R3
and finally
1 E(k)2 2 2 |ρ 1,0 (k)| dk ≤ 5 3 ||R||R 2 |k| 2 π
|l|≤
1 log dl ≤ 4 2 ||R||2R 2 E(2l)E(l) 2 π
by (55) which implies ||ρ1,0 ||C ≤
CR (log )1/2 ||Q||Q . 4π
4.3.2. Second order terms. To simplify the presentation, we introduce the following notation:
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
1 p du , q R3 E(u) +∞ dη 1 Kp := . 2π −∞ E(η)p
Sp,q := (4π)(2π)−3/p Sp := Sp,p ,
543
(56)
Let us recall the following inequality [48, Theorem 4.1]: ||f (x)g(−i∇)||Sp ≤ (2π)−3/p ||f ||Lp (R3 ) ||g||Lp (R3 ) , which implies
1 |D |a f 0
Sp
≤
(57)
Sp,ap ||f ||Lp (R3 ) . 4π
On the other hand, we shall often use the following trick (E(p)2 + η2 )(E(q)2 + η2 ) = (E(p)E(q))2 + (E(p)2 + E(q)2 )η2 + η4 1 1 ≥ E(p + q)2 + E(p + q)2 η2 4 2 1 ≥ E(p + q)2 E(η)2 4 by Lemma 6. This implies
1
E(p)2 + η2 E(q)2 + η2
≤
2 . E(p + q)E(η)
Recall now that we have Q2 = Q2,0 + Q1,1 + Q0,2 with +∞ 1 1 1 1 Q2,0 = − dη 0 R R , 2π −∞ D + iη D 0 + iη D 0 + iη +∞ 1 1 1 1 Q1,1 = dη 0 R ϕ 2π −∞ D + iη D 0 + iη D 0 + iη +∞ 1 1 1 1 + dη 0 ϕ R , 2π −∞ D + iη D 0 + iη D 0 + iη +∞ 1 1 1 1 Q0,2 = − dη 0 ϕ 0 ϕ 0 . 2π −∞ D + iη D + iη D + iη We shall now treat each term separately. Lemma 13. We have ρ0,2 = 0 and the following estimates: ||Q2,0 ||Q ≤ 25/2 K3/2 (CR )2 ||Q||2Q , √ ||Q0,2 ||Q ≤ 2 10S6,5 CM C6 ||ρ ||2C , ||ρ2,0 ||C ≤ and so
S6 C6 (CR )2 ||Q||2Q , π
||Q1,1 ||Q ≤ 4S6 C6 K3/2 CR ||Q||Q ||ρ ||C ,
||ρ1,1 ||C ≤ 4
S6,5 CM C6 CR ||ρ ||C ||Q||Q , π
2 ||(Q2 , ρ2 )|| ≤ κ2 (Q, ρ ) ,
(58)
´ S´er´e C. Hainzl, M. Lewin, E.
544
with √ √ κ2 = CQ2 CR 2 + 2 πCρ2 , √ S6 C6 K3/2 5S6,5 CM C6 3/2 CQ2 = max 2 K3/2 , √ , , √ 2π π 2 S6 C6 S6,5 CM C6 , Cρ2 = max , √ 2π π 3/2 2 where CM is a constant defined in Lemma 14. Proof. Step 1. Estimates on the exchange term Q2 . • Q2,0 . To estimate Q2,0 , we write |Q 2,0 (p, q)| ≤
1 2π
+∞
−∞
dη
p1 )| 1 , q)| |R(p, |R(p 1 dp1 , 2 2 2 2 3 E(p) + η E(p1 ) + η E(q)2 + η2 R
and so by (58) 3/2
2 |Q 2,0 (p, q)| ≤ 2π
+∞
−∞
p1 )| 1 , q)| dη |R(p, |R(p dp1 , E(η)3/2 E(p + q)1/2 R3 E(p + p1 )1/2 E(p1 + q)1/2
which implies E(p − q)E(p + q)1/2 |Q 2,0 (p, q)| p1 )| E(p1 − q)|R(p 1 , q)| E(p − p1 )|R(p, ≤ 25/2 K3/2 dp1 , E(p + p1 )1/2 E(p1 + q)1/2 R3 and finally ||Q2,0 ||Q ≤ 25/2 K3/2 ||R||2R ≤ 25/2 K3/2 (CR )2 ||Q||2Q . • Q1,1 . We treat for instance Q1,1 :=
1 (2π)5/2
+∞ −∞
dη
D0
1 1 1 R 0 ϕ 0 + iη D + iη D + iη
and use the same method to obtain (p, q)| E(p − q)E(p + q)1/2 |Q 1,1 +∞ 4 dη E(p − p1 ) ≤ |R(p, p1 )| 1/2 5/2 (2π) −∞ E(η) R3 E(p + p1 ) E(p1 − q) |ϕ (p1 − q)| dp1 . × (E(p1 )2 + η2 )1/4 (E(q)2 + η2 )1/4
This means that ||Q1,1 ||Q
4 ≤ 2π
+∞
−∞
dη 1 1 , f R 2 2 1/4 2 2 1/4 E(η) (|D0 | + η ) (|D0 | + η ) S2
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
545
where we have introduced R and f defined by (p, q) := E(p − q) |R(p, q)|, f(k) := E(k)|ϕ (k)|. R E(p + q)1/2 But now 1 1 1 1 f ≤ R f R 1 1 S2 (|D |2 +η2 ) 41 (|D |2 +η2 ) 41 2 2 2 2 4 4 (|D0 | + η ) (|D0 | + η ) S∞ 0 0 S2 1 1 ≤ ||R||R f . (|D |2 + η2 ) 41 (|D |2 + η2 ) 41 0 0 S6 If we now use inequality (57), we obtain 1 1 f ≤ 1 1 (|D |2 + η2 ) 4 (|D |2 + η2 ) 4 0
2 1 |f |1/2 1 (|D |2 + η2 ) 4 0 S6 S12 1/6 du ||f ||L6 ≤ (2π)−1/2 2 2 3 R3 1 + |u| + η ) S6 ||f ||L6 . = 4π E(η)1/2
0
Finally since ||f ||L6 ≤ C6 ||∇f ||L2 = C6 ||ϕ ||Y ,
(59)
and ||ϕ ||Y = (4π )||ρ ||C , we obtain ||Q1,1 ||Q
4S6 C6 ≤ 2π
+∞
−∞
dη ||R||R ||ρ ||C E(η)3/2
and ||Q1,1 ||Q ≤ 8S6 C6 K3/2 CR ||Q||Q ||ρ ||C . • Q0,2 . Unfortunately, the method used above cannot be applied to Q0,2 . In this case, we have to calculate this term explicitly. We can write 1 2 3 Q0,2 = Q0,2 , 1 ,2 ,3 ∈{±}
where for instance (by a residuum formula) +++ −−− Q 0,2 (p, q) = Q0,2 (p, q) = 0, + (p)ϕ (p − p1 )− (p1 )ϕ (p1 − q)− (q) +−− −3 Q (p, q) = (2π) , dp 1 0,2 (E(p) + E(q))(E(p) + E(p1 )) R3 + (p)ϕ (p − p1 )− (p1 )ϕ (p1 − q)+ (q) +−+ −3 Q (p, q) = (2π) dp1 , 0,2 (E(p) + E(p1 ))(E(q) + E(p1 )) R3
´ S´er´e C. Hainzl, M. Lewin, E.
546
+−− 1 2 3 and similar formulas for the other Q0,2 . We now treat for instance Q 0,2 . Using (30), we may obtain E(p − p1 ) +−− 1/2 −3 dp1 E(p − q)E(p + q) |Q0,2 | ≤ 2(2π) 3 E(p + p1 )2/3 R E(p1 − q)|ϕ (p1 − q)| × + (p)ϕ (p − p1 )− (p1 ) × dp1 . E(p1 )1/3 E(q)1/2
So, we may write Mf || ≤ 2 ||Q+−− Q 0,2
1 1 1 1 Mf f ≤ 2 f , S2 |D |1/3 |D |1/2 |D0 |1/3 |D0 |1/2 S2 0 0 S∞
where −3/2 M f (p, q) := (2π)
f(p − q) + (p)− (q) . E(p + q)2/3
Lemma 14. When Mf is defined by formula (60), then 1/2 2 2 Mf ≤ CM |k| |f (k)| dk , S2 4π R3
1/2 2 ∞ where CM := 2 0 E(2t)t 4/3dtE(t)2 2.1589. 2 Proof. We have + (p)− (q) = Tr C4 + (p)− (q) and so, by (54), |p − q|2 |f(p − q)|2 2 −3 dp dq dp dq|M f (p, q)| ≤ (2π) 2E((p + q)/2)2 E(p + q)4/3 du −3 2 2 dk|k| |f (k)| ≤ (2π) 2E(2u)4/3 E(u)2 ∞ t 2 dt −2 2 2 dk|k| |f (k)| ≤ (2π) . E(2t)4/3 E(t)2 0 Finally, since by (59) 1 1 1 1 S6,5 f ≤ f |D |1/3 |D |1/2 ≤ 4π ||f ||L6 , |D |1/3 |D |1/2 0 0 0 0 S∞ S6 we obtain 2 ||Q+−− 0,2 ||Q ≤ 2S6,5 CM C6 ||ρ ||C .
This result is immediately extended to the others terms and since we can prove |Q|2 = |Q++− + Q+−− |2 + |Q−++ + Q−−+ |2 + |Q+−+ |2 + |Q−+− |2 , we arrive at
√ ||Q0,2 ||Q ≤ 2 10S6,5 CM C6 ||ρ ||2C .
(60)
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
547
Step 2. Estimates on the density ρ2 . Let us now treat the density ρ2 . The general idea of the proof is to estimate ρ2 , ζ in terms of the norm ||ζ ||C by using | ρ, ζ | = |Tr (Qζ )| = Tr Qζ = Tr C4 | Qζ (p, p)|dp. Qζ (p, p) dp ≤ R3
R3
This can be done if we know that Qζ ∈ S1 . But we have 1 ≤ Q|D0 |2 1 ζ ||Qζ ||S1 = Q|D0 |2 ζ 2 2 S2 |D0 | S |D0 | S1 2 S2,4 2 ||ζ ||L2 , ≤ E() ||Q||S2 4π showing that Qζ ∈ S1 when ζ ∈ L2 . So, in what follows, we shall assume that ζ ∈ C ∩ L2 and prove a bound depending only on ||ζ ||C . By the density of C ∩ L2 in C , this will give us a bound on ||ρ||C . Let us remark first that ρ0,2 vanishes. Indeed we have 1 Tr C4 Q ρ 0,2 (k) = 0,2 (p + k/2, p − k/2) dp 3/2 (2π) |p|≤ and Tr C4 Q 0,2 (p, q) =
1 (2π)4
+∞ −∞
dη
dp1 Tr C4 R3
1 ϕ (p − p1 ) D0 (p) + iη
1 1 ϕ (p1 − q) × D0 (p1 ) + iη D0 (q) + iη +∞ 1 ϕ (p − p1 )ϕ (p1 − q) = dη dp1 4 (2π ) −∞ E(p)2 + η2 E(p1 )2 + η2 E(q)2 + η2 R3 ×Tr C4 (D0 (p) − iη))(D0 (p1 ) − iη)(D0 (q) − iη) . Now the terms linear in the Dirac matrices are traceless and the remaining terms are odd in η and vanish after integration. This can be easily generalized to ρ0,2k for all k, and is known as Furry’s Theorem in the physics literature [22]. • ρ2,0 . We use here a method similar to what we have done above. We estimate for some ζ ∈ C ∩ L2 and Qζ := Q2,0 ζ , +∞ 1 , p2 )| | |R(p, p1 )| |R(p ζ (p2 − p)| dp1 dp2 −5/2 dη , |Qζ (p, p)| ≤ (2π ) 2 2 2 E(p) + η E(p1 ) + η2 E(p2 )2 + η2 −∞ +∞ dη ≤ 4(2π)−5/2 E(η) −∞ p1 )| E(p1 − p2 )|R(p 1 , p2 )| E(p − p1 )|R(p, × dp1 dp2 E(p + p1 )1/2 E(p1 + p2 )1/2 | ζ (p2 − p)| × , 2 E(p2 − p)(E(p) + η2 )1/4 (E(p2 )2 + η2 )1/4 +∞ dη R (p, p1 )R (p1 , p2 )ζ (p2 − p) ≤ 4(2π)−5/2 dp1 dp2 , (E(p)2 + η2 )1/4 (E(p2 )2 + η2 )1/4 −∞ E(η)
´ S´er´e C. Hainzl, M. Lewin, E.
548
(p, q) = where R
E(p−q)|R(p,q)| E(p+q)1/2
ζ (k). This means that and ζ (k) = E(k)−1
dη 1 1 ζ RR 2 2 1/4 2 2 1/4 (|D0 | + η ) (|D0 | + η ) S1 −∞ E(η) +∞ S6 C6 (CR )2 K3/2 dη 2 4S6 R ζ ≤ ≤ ||Q||2Q ||ζ ||C 6 S L 2 (2π )(4π) −∞ E(η)3/2 π
4 | ρ2,0 , ζ | ≤ 2π
+∞
by (59), showing that ||ρ2,0 ||C ≤
S6 C6 (CR )2 K3/2 ||Q||2Q . π
• ρ1,1 . Unfortunately, as for Q0,2 , we have to calculate ρ1,1 explicitly. Let us start for +−− instance with ρ1,1 , the density associated with one of the two terms of Q1,1 , (2π )−3/2
R3
dp1
− p1 )− (p1 )ϕ (p1 − q)− (q) + (p)R(p . (E(p) + E(q))(E(p) + E(p1 ))
We use the same method as above and estimate for some ζ ∈ C ∩ L2 the term −3 Qζ (p, p) = (2π) dp1 dp2 ×
p1 )− (p1 )ϕ (p1 − p2 )− (p2 ) + (p)R(p, ζ (p2 − p), (E(p) + E(p2 ))(E(p) + E(p1 ))
by
p1 )− (p1 )| |ϕ (p1 − p2 )| |+ (p)R(p, E(p + p1 )1/2 E(p1 )1/2 E(p2 )1/3 − + |ζ (p2 − p)| × | (p2 ) (p)| dp1 dp2 , × E(p + p2 )2/3 p1 )− (p1 )| E(p − p1 )|+ (p)R(p, ≤ 2(2π)−3/2 E(p + p1 )1/2 f(p1 − p2 ) × M ζ (p2 , p)dp1 dp2 , E(p1 )1/2 E(p2 )1/3
ζ (p, p)| ≤ (2π)−3 |Q
ζ (k)|/E(k). Now with f(k) := E(k)|ϕ (k)| and ζ (k) := | ζ (p, p)| ≤ 2(2π)−3/2 |Q
1 (p, p1 ) R
f(p1 − p2 ) M ζ (p2 , p) dp1 dp2 E(p1 )1/2 E(p2 )1/3
with + − 1 (p, p1 ) := E(p − p1 )| (p)R(p, p1 ) (p1 )| . R 1/2 E(p + p1 )
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
We thus have +−− | ρ1,1 , ζ |
549
1 1 ≤ 2 R1 f Mζ |D0 |1/2 |D0 |1/3 1 S 1 1 + − ≤ 2|| R ||R Mζ S f 2 |D |1/2 |D0 |1/3 S∞ 0 1 1 ≤ 2||+ R− ||R Mζ S f 2 |D |1/2 |D |1/3 0
0
S6
S6,5 CM C6 + ≤ || R− ||R ||ζ ||C ||ρ ||C , 2π and finally +−− ||ρ1,1 ||C ≤
S6,5 CM C6 + || R− ||R ||ρ ||C . 2π
+−+ We now treat ρ1,1 and estimate −3 Qζ (p, p) = (2π) dp1 dp2
×
p1 )− (p1 )ϕ (p1 − p2 )+ (p2 ) + (p)R(p, ζ (p2 − p), (E(p) + E(p1 ))(E(p1 ) + E(p2 ))
by
p1 )− (p1 )| E(p − p1 )|+ (p)R(p, E(p + p1 )1/2 f(p1 − p2 )|− (p1 )+ (p2 )| ζ (p2 − p) × dp1 dp2 . E(p1 + p2 )2/3 E(p2 )5/6
ζ (p, p)| ≤ 2(2π)−3 |Q
Using the same argument as above, we arrive at +−+ ||C ≤ ||ρ1,1
S6,5 CM C6 + || R− ||R ||ρ ||C . 2π
++− , we remark that To treat ρ1,1
1 1 ≤ (E(p) + E(p2 ))(E(p1 ) + E(p2 )) (E(p) + E(p1 ))(E(p1 ) + E(p2 )) 1 + (E(p) + E(p1 ))(E(p) + E(p2 )) and use the same estimates as above to get ++− ||ρ1,1 ||C ≤
= ||R||2R , we end up with S6,5 CM C6 CR S6,5 CM C6 ||ρ ||C ||ρ ||C ||Q||Q . ||1 R2 ||R ≤ 4 ||ρ1,1 ||C ≤ 2 π π Finally, since
1 ,2 ∈{±} ||
S6,5 CM C6 + || R− ||R ||ρ ||C . π
1 R2 ||2
R
1 ,2 ∈{±}
´ S´er´e C. Hainzl, M. Lewin, E.
550
4.3.3. The general nth order case. Now that we have explained how the proof works for the second order, let us estimate the general nth order term. Lemma 15. We have the following estimates √ S6 C6 3 n ∀n ≥ 3, ||Qn ||Q ≤ nK n2 CQ (Q, ρ ) , with CQ = 2 √ , 2 π n S6 C6 S6 C6 5 ∀n ≥ 5, ||ρn ||C ≤ nK n+1 Cρ (Q, ρ ) , with Cρ = , √ 2 4π 2 π K2 S6 C6 S6 C6 2 4 . ||ρ4 ||C ≤ Cρ4 (Q, ρ ) , with Cρ4 := √ π 2 π Therefore,
n ||(Qn , ρn )|| ≤ κn (Q, ρ )
∀n ≥ 4, with
√ √ κ4 = 4CR K2 CQ 2 + 2 πCρ4 ,
√ √ κn = nCR K n2 CQ 2 + 2nK n+1 Cρ π . 2
Remark that it can be proved that Kn ∼n→∞ √Cn , which gives the claimed behaviour for κn as n → ∞. Proof. Step 1. Estimates on the exchange term Qn . • Qk,l with k ≥ 1 and k + l = n ≥ 3. Recall that Qk,l
(−1)l+1 = 2π
+∞
I ∪J ={1,...,n}, |I |=k, |J |=l −∞
n 1 1 Rj 0 dη 0 , D + iη D + iη j =1
where Rj = R if j ∈ I and Rj = ϕ if j ∈ J . For the sake of simplicity, we treat only k l +∞ 1 1 1 1 Qk,l = ϕ 0 R 0 dη 0 . 2π −∞ D + iη D + iη D + iη We have (p, q)| ≤ |Q k,l
1 (2π )
1+ 3l2
+∞ −∞
dη
···
1 p1 )| |R(p, (E(p)2 + η2 )1/2
1 × (E(p1 )2 + η2 )1/4 k−1 1 1 × |R(pj , pj +1 )| (E(pj )2 + η2 )1/4 (E(pj +1 )2 + η2 )1/4 j =1
×
n−2 j =k
×
1 1 |ϕ (pj − pj +1 )| 2 2 1/4 (E(pj ) + η ) (E(pj +1 )2 + η2 )1/4
1 1 |ϕ (pn−1 − q)| dp1 · · · dpn−1 , 2 2 1/4 2 (E(pn−1 ) + η ) (E(q) + η2 )1/2
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
551
so by (58), (p, q)| E(p − q)E(p + q)1/2 |Q k,l k+1 +∞ p1 )| |R(p, dη 2 2 E(p − q) ··· ≤ 3l k+1 E(p + p1 )1/2 −∞ E(η) 2 (2π )1+ 2
×
k−1
j , pj +1 )| |R(p E(pj + pj +1 )1/2
j =1
×
n−2 j =k
×
1 1 |ϕ (pj − pj +1 )| 2 2 1/4 (E(pj ) + η ) (E(pj +1 )2 + η2 )1/4
1 1 |ϕ (pn−1 − q)| dp1 · · · dpn−1 . 2 2 1/4 2 (E(pn−1 ) + η ) (E(q) + η2 )1/4
Now if we use the easy generalization of (28), E(p − q) ≤ E(p − p1 ) + E(p1 − p2 ) + · · · + E(pn−2 − pn−1 ) + E(pn−1 − q), we obtain by a similar argument as before $ k+1 ∞ 2 2 dη k(CR )k (4πC∞ )l ||Qk,l ||Q ≤ k+1 2π −∞ E(η)l+ 2 % ∞ dη k l−1 +l(CR ) (4πC∞ ) S6 C6 ||Q||kQ ||ρ ||lC . k −∞ E(η)l+ 2 To obtain this result, we have estimated each term containing a ϕ by using 1 1 1 C∞ ϕ L∞ ≤ ||ϕ ||Y ≤ (|D |2 + E(η)2 )1/4 ϕ (|D |2 + E(η)2 )1/4 E(η) E(η) 0 0 S∞ and when E(pj − pj +1 ) appears in front of a ϕ (pj − pj +1 ) (i.e. when j ≥ k), by using 1 1 (|D |2 + E(η)2 )1/4 f (|D |2 + E(η)2 )1/4 0 0 S∞ 1 1 S6 ≤ ||f ||L6 . ≤ f 2 2 1/4 2 2 1/4 (|D | + E(η) ) (|D | + E(η) ) E(η)1/2 4π 0
So we have
0
S6
' k(CR )k (4πC∞ )l Kl+ k+1 +l(CR )k (4π C∞ )l−1 S6 C6 Kl+ k ||Q||kQ ||ρ ||lC 2 2 k+1 S 6 C6 k l k l ||Q||Q ||ρ ||C ≤ 2 2 n(CR ) (4πC∞ ) Kl+ k max 1, 2 4π C∞
||Qk,l ||Q ≤ 2
k+1 2
&
which implies ||Qk,l ||Q ≤
n k
2
k+1 2
S 6 C6 ||Q||kQ ||ρ ||lC . n(CR )k (4πC∞ )l Kl+ k max 1, 2 4π C∞
´ S´er´e C. Hainzl, M. Lewin, E.
552
• Q0,n with n ≥ 3. Recall that Q0,n =
(−1)n+1 2π
+∞
−∞
dη
n 1 1 ϕ D 0 + iη D 0 + iη
so that E(p − q)E(p + q)1/2 |Q 0,n (p, q)| ≤ ×
(E(p)2 n−2
×
j =1
×
E(p − q) 3n
(2π)1+ 2
+∞ −∞
dη 1
E(η) 2
· · · dp1 · · · dpn−1
|ϕ (p − p1 )| + η2 )1/4 (E(p1 )2 + η2 )1/4
1 1 |ϕ (pj − pj +1 )| 2 2 1/4 (E(pj ) + η ) (E(pj +1 )2 + η2 )1/4
1 1 |ϕ (pn−1 − q)| . (E(pn−1 )2 + η2 )1/4 (E(q)2 + η2 )1/4
We now use (59) to bound for some f1 , f2 and f3 , 3 1 1 fj 2 2 1/4 2 2 1/4 (|D0 | + η ) j =1 (|D0 | + η )
≤ S2
3 (S6 )3 fj 6 L 3/2 3 E(η) (4π )
(61)
j =1
to obtain √ ∞ dη 2 (S6 C6 )3 (4π C∞ )n−3 ||ρ ||nC ||Q0,n ||Q ≤ n 2+(n−3) 2π −∞ E(η) or √ ||Q0,n ||Q ≤ nKn−1 2(S6 C6 )3 (4πC∞ )n−3 ||ρ ||nC . √ Finally, we can write for instance (recall that C∞ = 1/(2 π )) √
n √ ||Qn ||Q ≤ nKn/2 CQ CR 2||Q||Q + 2 π ||ρ ||C , where √ (S6 C6 )3 √ S 6 C6 S6 C6 3 CQ = 2 max 1, = 2 , , 4πC∞ 4πC∞ 8π 3/2 and since Kn/2 ≥ Kn−1 when n ≥ 2. Step 2. Estimates on the density ρn .
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553
of the Q • ρk,l with k ≥ 2 and n ≥ 3. As before we treat for instance the density ρk,l k,l where the k R’s are on the left and the l ϕ ’s are on the right. For some fixed ζ ∈ C ∩ L2 , we introduce Qζ := Qk,l ζ . We thus estimate
1
ζ (p, p)| ≤ |Q (2π)
1+ 3(l+1) 2
+∞ −∞
dη
···
1 p1 )| |R(p, (E(p)2 + η2 )1/4
1 × 2 (E(p1 ) + η2 )1/4 k−1 1 1 j , pj +1 )| × | R(p (E(pj )2 + η2 )1/4 (E(pj +1 )2 + η2 )1/4 j =1
×
n−1 j =k
×
1 1 |ϕ (pj − pj +1 )| (E(pj )2 + η2 )1/4 (E(pj +1 )2 + η2 )1/4
1 1 | ζ (pn − p)| dp1 · · · dpn . (E(pn )2 + η2 )1/4 (E(p)2 + η2 )1/4
We now use as before | ζ (pn − p)| ≤
| ζ (pn − p)| E(p − p1 ) + E(p1 − p2 ) + · · · + E(pn−1 − pn ) E(pn − p)
to obtain | ρk,l , ζ |
2k/2 S6 C6 ≤n (CR )k (4πC∞ )l 4π
1 2π
∞
−∞
dη E(η)
l+ k+1 2
||Q||kQ ||ρ ||lC ||ζ ||C ,
and so ||ρk,l ||C ≤ n
n S6 C6 k
4π
√ (CR 2)k (4πC∞ )l Kl+ k+1 ||Q||kQ ||ρ ||lC . 2
• ρ1,l with l ≥ 2. We may treat for instance with the same notation as before +∞ 1 1 ζ (p, p)| ≤ p1 )| |Q dη · · · |R(p, 2 + η2 )1/4 1+ 3n (E(p) (2π) 2 −∞ 1 × (E(p1 )2 + η2 )1/4 n−1 1 1 × |ϕ (pj − pj +1 )| (E(pj )2 + η2 )1/4 (E(pj +1 )2 + η2 )1/4 j =1
×
(E(pn
1 1 | ζ (pn − p)| dp1 · · · dpn . 2 2 1/4 +η ) (E(p) + η2 )1/4
)2
We now use (61) and obtain | ρ1,l , ζ |
21/2 (S6 C6 )3 CR (4πC∞ )l−2 ≤n 4π
1 2π
∞
−∞
dη E(η)
l−2+ 24
||Q||Q ||ρ ||lC ||ζ ||C
´ S´er´e C. Hainzl, M. Lewin, E.
554
and so ||ρ1,l ||C ≤ n
n
21/2 (S6 C6 )3 Kl
n−1
4π
CR (4πC∞ )l−2 ||Q||Q ||ρ ||lC .
• ρ0,l with l ≥ 5. We want to estimate 1
ζ (p, p)| ≤ |Q (2π )
1+ 3(n+1) 2
+∞ −∞
dη
···
(E(p)2
1 |ϕ (p − p1 )| + η2 )1/4
1 (E(p1 )2 + η2 )1/4 n−1 1 1 (pj − pj +1 )| × | ϕ (E(pj )2 + η2 )1/4 (E(pj +1 )2 + η2 )1/4 ×
j =1
×
1 1 | ζ (pn − p)| dp1 · · · dpn . (E(pn )2 + η2 )1/4 (E(p)2 + η2 )1/4
Since there are at least 6 functions, we may use (61) twice and obtain | ρ0,l , ζ |
(S6 C6 )6 ≤n (4πC∞ )l−5 4π
1 2π
∞
−∞
dη E(η)l−2
||ρ ||nC ||ζ ||C ,
and so ||ρ0,l ||C ≤ n
(S6 C6 )6 (4πC∞ )l−5 Kl−2 ||ρ ||nC . 4π
Now since Kn−2 ≤ K(n+1)/2 when n ≥ 5, we obtain √
n √ ||ρn ||C ≤ nK n+1 Cρ CR 2||Q||Q + 2 π ||ρ ||C 2
(63)
with S6 C6 C6 S6 5 C6 S6 2 S6 C6 C6 S6 5 Cρ := = max 1, √ , √ . √ 4π 4π 2 π 2 π 2 π For ρ4 , we notice that ρ0,4 = 0 for the same reason as ρ0,2 , and that K2 ≥ K5/2 . Therefore we obtain √
4 √ ||ρ4 ||C ≤ Cρ4 CR 2||Q||Q + 2 π ||ρ ||C (64) with Cρ4
S6 C6 K2 C6 S6 2 4K2 S6 C6 C6 S6 2 = := . max 1, √ √ 4π π 2 π 2 π
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
555
4.3.4. The third order density ρ3 . Lemma 16. We have
√
3 √ ||ρ3 ||C ≤ Cρ3 CR 2||Q||Q + 2 π ||ρ ||C
and therefore
3 ||(Q3 , ρ3 )|| ≤ κ3 (Q, ρ )
with √ √ κ3 = 3CR K3/2 CQ 2 + 2 πCρ3 ,
15CM S6 (S6,4 )2 (C6 )4 . π(4π C∞ )3 Proof. Notice that thanks to the previous proof, we already have some estimates on ρ3,0 , ρ2,1 and ρ1,2 . It remains to study ρ0,3 . As before and as in [29], we have to compute ρ0,3 explicitly by a residuum formula. We thus write 1 2 3 4 ρ0,3 ρ0,3 = Cρ3 =
1 ,...,4 ∈{±}
with an obvious definition. +−−− • Let us treat first ρ0,3 . We thus fix some ζ ∈ C ∩ L2 and estimate the term + (p)ϕ (p − p1 )− (p1 ) ζ (p, p) = (2π)−6 Q dp1 dp2 dp3 E(p) + E(p1 ) − ϕ (p1 − p2 ) (p2 )ϕ (p2 − p3 )− (p3 ) × ζ (p3 − p), (E(p) + E(p2 ))(E(p) + E(p3 )) by + (p)ϕ (p − p1 )− (p1 ) −6 ζ (p, p)| ≤ (2π) dp1 dp2 dp3 |Q E(p + p1 )2/3 ϕ (p2 − p3 ) ϕ (p1 − p2 ) . ζ (p × − p) 3 E(p1 )1/3 E(p2 )E(p) So if we follow the method used above, we obtain +−−− ||ρ0,3 ||C ≤
3CM S6 (S6,4 )2 (C6 )4 3 ||ρ ||C . 4π
−+++ −−−+ +++− −+−− +−++ −−+− Now, it is easily seen that ρ0,3 , ρ0,3 , ρ0,3 , ρ0,3 , ρ0,3 , ρ0,3 and ++−+ ρ0,3 can be treated by exactly the same method. ++−− • Let us now treat for instance ρ0,3 . Thanks to the residuum formula, we have to study −6 dp1 dp2 dp3 + (p)ϕ (p − p1 )+ (p1 )ϕ (p1 − p2 )− (p2 ) Qζ (p, p) = (2π )
ϕ (p2 − p3 )− (p3 ) ζ (p3 − p) 1 × (E(p) + E(p2 ))(E(p1 ) + E(p2 ))(E(p1 ) + E(p3 )) 1 + . (E(p) + E(p2 ))(E(p) + E(p3 ))(E(p1 ) + E(p3 ))
´ S´er´e C. Hainzl, M. Lewin, E.
556
Table 1. Constants used in the proof of Theorem 3 Constant C6 , C∞ CR κ1 () Sp,q , Sp , Kp κ2 , CQ2 , Cρ2 CM CQ , Cρ , Cρ4 κn , n ≥ 4 Cρ3 , κ3
defined in Lemma 3 Equation (34) in Lemma 8 Equation (52) in Proposition 10 Equation (56) Lemma 13 Lemma 14 Lemma 15 Lemma 15 Lemma 16
If we now use the same method as above for each of the two terms of this sum, we arrive at ++−− ||ρ0,3 ||C ≤ 2
3CM S6 (S6,4 )2 (C6 )4 3 ||ρ ||C . 4π
−−++ +−+− −+−+ +−−+ −++− This is easily generalized to the study of ρ0,3 , ρ0,3 , ρ0,3 , ρ0,3 and ρ0,3 . Summing now all these terms, we obtain
||ρ0,3 ||C ≤ 20
3CM S6 (S6,4 )2 (C6 )4 3 15CM S6 (S6,4 )2 (C6 )4 3 ||ρ ||C = ||ρ ||C 4π π
and √
3 √ ||ρ3 ||C ≤ Cρ3 CR 2||Q||Q + 2 π ||ρ ||C ,
(65)
with Cρ3
S6 C6 2 20CM (S6,4 )2 (C6 )3 S 6 C6 max K3+1/2 , K3 , K2 =3 , 4π 4πC∞ (4π C∞ )3 =
15CM S6 (S6,4 )2 (C6 )4 . π(4πC∞ )3
4.3.5. List of constants. We have used many constants in this proof. A summary which should help the reader to follow our arguments is provided in Table 1. Appendix. Derivation of the BDF Energy In this section, we recall some basics about the second-quantization in no-photon QED and explain how the BDF energy E is derived from this theory, as a mean-field approximation. We mainly follow the method of Chaix-Iracane [9, 11], but with the notation of [54, 4, 30]. See also [29] for more details concerning the polarization of the vacuum. To simplify the presentation, we introduce P−0 := P 0 and P+0 := 1 − P 0 .
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
557 (1)
(1)
Free particles, Fock space, free vacuum. We first introduce F+ := P+0 H and F− := CP−0 H which are called respectively the free electron and the free positron state sub(0) space. C is the charge-conjugation operator defined by Cψ = iβα2 ψ. We define F+ = (n (m (0) (n) (1) (m) (1) F− = C and F+ = k=1 F+ , F− = k=1 F− for n, m ≥ 1. The space of n (n) (m) free electrons and m free positrons is then defined by F (n,m) = F+ ⊗ F− and the associated Fock space is ∞ )
F :=
F (n,m) .
(66)
n,m=0
For any f ∈ H , the free electron (resp. positron) annihilation and creation operators a0 (f ) and a0∗ (f ) (resp. b0 (f ) and b0∗ (f )) are defined as usually [54]. They fulfill the Canonical Anti-commutation Relations {a0 (f ), a0 (g)} = {a0 (f ), b0 (g)} = {b0 (f ), b0 (g)} = 0, {a0 (f ), a0∗ (g)} = f, P+0 g , {b0∗ (f ), b0 (g)} = f, P−0 g ,
(67) (68)
where ·, · denotes the usual scalar product of L2 (R3 , C4 ). The free vacuum state 0 , a unit vector spanning F (0,0) = C, is uniquely characterized up to a phase factor by the properties ||0 ||F = 1,
a0 (f )0 = 0
b0 (f )0 = 0,
and
(69)
for all f ∈ H . The field operator (f ) is defined on the Fock space F by (f ) = a0 (f ) + b0∗ (f ). 2, In terms of (f ), the CAR become, for all (f, g) ∈ H
{(f ), (g)} = { ∗ (f ), ∗ (g)} = 0,
{(f ), ∗ (g)} = f, g 1.
Dressed particles and vacuum. In this description, the free electrons and positrons are 0 ⊕ H0 . defined with respect to the projector P 0 , or equivalently the splitting H = H− + We want now to change this definition and introduce the dressed electrons and positrons. To this end, we fix a new projector P on H , use again the notation P− := P and P+ := 1 − P , and introduce the dressed particle annihilation operators aP (f ) := (P+ f ),
bP (f ) := ∗ (P− f ).
(70)
A similar formula can be given for the dressed particle creation operators aP∗ and bP∗ . These dressed operators satisfy the same CAR as for the free operators (67), (68). We also introduce the dressed electrons and positrons state subspaces P H+ = (1 − P )H ,
P H− = P H .
Now, the main question is to know if there exists a dressed vacuum P in the Fock space F. This state has to be a solution to the analogue of (69), ||P ||F = 1,
aP (f )P = 0
and
for all f ∈ H . The answer is given by the celebrated
bP (f )P = 0
(71)
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558
Theorem 4 (Shale-Stinespring Theorem [52]). There exists a dressed vacuum P in the Fock space F satisfying (71) if and only if P − P 0 is a Hilbert-Schmidt operator. In this case, P is unique up to a phase factor. There are many proofs of this theorem in the literature, see, e.g., [54, 36, 47] and the references in [20]. This result explains why we assumed in the previous section that P − P 0 ∈ S2 (H ). Notice that P can be expressed as a rotation of the bare vacuum in the Fock space, P = U0 , U being called a Bogoliubov transformation. An explicit formula for P can be found in a lot of papers [54, 36, 47, 49, 50, 20]. The charge of the dressed vacuum P can be easily computed2
P , Q P = tr[P+0 (P − P 0 )P+0 ] + tr[P−0 (P − P 0 )P−0 ] = tr P 0 (P − P 0 ), justifying our study of Sect. 2.1. Here Q is the charge operator defined on F by [54, Formula (10.52)] * Q= a0∗ (fi )a0 (fi ) − b0∗ (fi )b0 (fi ) = a0∗ (fi )a0 (fi ) − b0∗ (fi )b0 (fi ), i∈Z\{0}
i≥1
i≤−1
0 and (f ) where (fi )i≥1 is an orthonormal basis of H+ i i≤−1 is an orthonormal basis of 0 H− .
Second-quantized Hamiltonian. In the physics literature, the creation and annihilation operators are defined differently. For instance, instead of a0∗ (f ) which creates an electron in the state P+0 f , the operator a0∗ (x) which creates an electron at x is formally ∗ 0 used, where a0∗ (x) = ∞ i=1 a0 (fi )fi (x), (fi )i≥1 being an orthonormal basis of H+ . The operators aP (x) and bP (x) are defined similarly. We shall now use this formalism which is also the one of [29]. Formally, the CAR (67, 68) are equivalent to {a0 (x), a0 (y)} = {a0 (x), b0 (y)} = {a0∗ (x), b0 (y)} = {b0 (x), b0 (y)} = 0,
(72)
{a0 (x), a0∗ (y)}
(73)
=
P+0 (x, y),
{b0∗ (x), b0 (y)}
= P (x, y). 0
We now start with writing down the formal unregularized no-photon Hamiltonian, ∗ (x)(x) ∗ (y)(y) α Hur = dx ∗ (x)D αϕ (x) + dx dy , (74) 2 |x − y| which acts on the Fock space F. As explained for instance in [54], the free vacuum may not belong to the domain of this formally defined operator. Therefore, the expression (74) is renormalized by using a procedure which is called “normal ordering”, denoted by double dots : − :P 0 . In each product of annihilation and creation operators, the a0∗ and b0∗ are moved to the left as if they anticommute with the a0 and b0 . For instance : ∗ (x)(y) :P 0 = a0∗ (x)a0 (y) + a0∗ (y)b0 (x) + b0 (x)a0 (y) − b0∗ (y)b0 (x) = ∗ (x)(y) − P 0 (x, y).
(75)
2 Notice that the charge of the dressed vacuum can also be easily obtained by using the explicit formula of P , which immediately shows that it is an integer [54, 37, 50, 49].
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
559
As a first step we thus regularize Hur as done in Chaix and Iracane [9, Sects. 3.5 and 4.1], namely we normal order with respect to the free projector P 0 , H=
dx : ∗ (x)D αϕ (x) :P 0 +
α 2
dx
dy
: ∗ (x)(x) ∗ (y)(y) :P 0 . (76) |x − y|
This kind of regularization, which follows ideas of Dirac [13, 14], is standard in QED. It corresponds to the subtraction of the energy of the free Dirac sea, and the interaction energy with the free Dirac sea. A physical justification is given in [29, Sect. 3], on the basis of two guiding principles formulated by Weisskopf in [56]. The same choice is made in other studies dealing with vacuum polarization, for instance [9, 11, 36, 37, 49, 50, 20, 1]. However, a better choice might be possible (see the paper [39] by Lieb and Siedentop, who propose another translation-invariant reference for normal ordering, in the absence of external field). Now we can express H in terms of : − :P for some other P . Using (75) we obtain the reordering relations : ∗ (x)(y) :P 0 =: ∗ (x)(y) :P +Q(x, y)
(77)
and : ∗ (x)(x) ∗ (y)(y) :P 0 =: ∗ (x)(x) ∗ (y)(y) :P +2 : ∗ (x)(x) :P Tr C4 (Q(y, y)) −2 : ∗ (x)(y) :P Q(x, y) +Tr C4 (Q(x, x))Tr C4 (Q(y, y)) − |Q(x, y)|2 , where Q = P − P 0 . Therefore we can rewrite H with respect to an arbitrary dressed vacuum P [9, formula (4.3)],
H=
α : ∗ (x)(x) ∗ (y)(y) :P : ∗ (x)D αϕ (x) :P dx + dx dy 2 |x − y| : ∗ (x)(x) :P Tr C4 (Q(y, y)) +α dx dy |x − y| : ∗ (x)(y) :P Q(x, y) −α dx dy |x − y| α Tr C4 (Q(x, x))Tr C4 (Q(y, y)) + tr P 0 (D ϕ Q) + dx dy 2 |x − y| α |Q(x, y)|2 − dx dy. (78) 2 |x − y|
The last two lines represent the energy of the dressed vacuum P measured with respect to P 0 , whereas in the second and third lines the vacuum polarization potentials appear.
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560
Restriction to Bogoliubov-Dirac-Fock states. We now follow [9] and restrict ourselves to Bogoliubov-Dirac-Fock type states. In this approximation method, a dressed vacuum P is first chosen such that P − P 0 ∈ S2 (H). Then a BDF state is simply a Slater determinant made with n electrons and m positrons defined with respect to the dressed vacuum P (n, m ≥ 0 are not fixed in this theory). This is a state of F which takes the form ψ = aP∗ (f1 ) · · · aP∗ (fn ) bP∗ (g1 ) · · · bP∗ (gm ) P , P )n and (g , ..., g ) ∈ (HP )m are such that f , f = δ , where (f1 , ..., fn ) ∈ (H+ 1 m i j ij −
gi , gj = δij , and P is the dressed vacuum in F obtained by Theorem 4. Since it is easily seen that ψ = P , where
P = P + γ,
γ =
n i=1
|fi fi | −
m
|gj gj |,
j =1
we obtain immediately from (78),
ψ|H|ψ = P |H|P = E(P − P 0 ) = E(P − P 0 + γ ). In [9, Formula (4.8)], this formula is expanded like in (12) in terms of the vacuum density matrix Q = P − P 0 and the density γ of the dressed particles. Acknowledgement. C.H. wishes to thank Heinz Siedentop for suggesting him the possibility of studying a self-consistent model of the polarized vacuum, during his time as post-doc at the LMU (Munich). The authors acknowledge support from the European Union’s IHP network Analysis and Quantum HPRNCT-2002-00277. E.S. acknowledges support from the Institut Universitaire de France.
References 1. Aste, A., Baur, G., Hencken, K., Trautmann, D., Scharf, G.: Electron-positron pair production in the external electromagnetic field of colliding relativistic heavy ions. Eur. Phys. J. C 23(3), 545–550 (2002) 2. Avron, J., Seiler, R., Simon, B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994) 3. Bach, V.: Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147, 527–548 (1992) 4. Bach, V., Barbaroux, J.-M., Helffer, B., Siedentop, H.: On the stability of the relativistic electronpositron field. Commun. Math. Phys. 201, 445–460 (1999) 5. Bach, V., Lieb, E.H., Solovej, J.P.: Generalized Hartree-Fock theory and the Hubbard model. J. Statist. Phys. 76(1–2), 3–89 (1994) 6. Barbaroux, J.M., Esteban, M.J., S´er´e, E.: Some connections between Dirac-Fock and Electron-Positron Hartree-Fock. Ann. Henri Poincar´e 6(1), 85–102 (2005) 7. Barbaroux, J.-M., Farkas, W., Helffer, B., Siedentop, H.: On the Hartree-Fock equations of the electron/positron field. Commun. Math. Phys. 255, 131–159 (2005) 8. Canc`es, E.: SCF algorithms for HF electronic calculations. Defranceschi, M. et al. (ed.), Mathematical models and methods for ab initio quantum chemistry. Lect. Notes Chem. 74, Berlin: Springer, 2000, pp. 17–43 9. Chaix, P., Iracane, D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism. J. Phys. B 22(23), 3791–3814 (1989) 10. Chaix, P., Iracane, D., Lions, P.L.: From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation. J. Phys. B 22(23), 3815–3828 (1989) 11. Chaix, P.: Une M´ethode de Champ Moyen Relativiste et Application a` l’Etude du Vide de l’Electrodynamique Quantique. PhD Thesis, University Paris VI, 1990
Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation
561
12. Desclaux, J.P.: Relativistic Dirac-Fock expectation values for atoms with Z = 1 to Z = 120. Atomic Data and Nuclear Data Tables 12, 311–406 (1973) 13. Dirac, P.A.M.: Th´eorie du positron. Solvay report, Paris: Gauthier-Villars, XXV, 353 S., 1934, pp. 203–212 14. Dirac, P.A.M.: Discussion of the infinite distribution of electrons in the theory of the positron. Proc. Camb. Philos. Soc. 30, 150–163 (1934) 15. Esteban, M.J., S´er´e, E.: Solutions of the Dirac-Fock Equations for atoms and molecules. Commun. Math. Phys. 203, 499–530 (1999) 16. Esteban, M.J., S´er´e, E.: Nonrelativistic limit of the Dirac-Fock equations. Ann. Henri Poincar´e 2(5), 941–961 (2001) 17. Esteban, M.J., S´er´e, E.: A max-min principle for the ground state of the Dirac-Fock functional. Contemp. Math. 307, 135–139 (2002) 18. Foldy, L.L., Eriksen, E.: Some physical consequences of vacuum polarization. Phys. Rev. 95(4), 1048–1051 (1954) 19. French, J.D., Weisskopf, V.F.: The electromagnetic shift of energy levels. Phys. Rev., II. Ser. 75, 1240–1248 (1949) 20. Fierz, H., Scharf, G.: Particle interpretation for external field problems in QED. Helv. Phys. Acta 52, 437–453 (1979) 21. Furry, W.H.: On bound states and scattering in positron theory. Phys. Rev. 81(1), 115–124 (1951) 22. Furry, W.H.: A symmetry theorem in the positron theory. Phys. Rev. 51, 125–129 (1937) 23. Furry, W.H., Oppenheimer, J.R.: On the theory of the electron and positive. Phys. Rev., II. Ser. 45, 245–262 (1934) 24. Glauber, R., Rarita, W., Schwed, P.: Vacuum polarization effects on energy levels in µ-mesonic atoms. Phys. Rev. 120(2), 609–613 (1960) 25. Gorceix, O. Indelicato, P., Desclaux, J.P.: Multiconfiguration Dirac-Fock studies of two-electron ions: I. Electron-electron interaction. J. Phys. B: At. Mol. Phys. 20, 639–649 (1987) 26. Grant, I.P.: Relativistic calculation of atomic structures. Adv. Phys. 19, 747–811 (1970) 27. Hainzl, C.: On the vacuum polarization density caused by an external field. Ann. Henri Poincar´e 5, 1137–1157 (2004) 28. Hainzl, C., Lewin, M., S´er´e, E.: Self-consistent solution for the polarized vacuum in a no-photon QED model. To appear in J. Phys. A: Math. and Gen. 29. Hainzl, C., Siedentop, H.: Non-perturbative mass and charge renormalization in relativistic no-photon quantum electrodynamics. Commun. Math. Phys. 243, 241–260 (2003) 30. Helffer, B., Siedentop, H.: Form perturbation of the second quantized Dirac field. Mathematical Physics Electronic Journal 4, paper 4, 1998 31. Heisenberg, W.: Bemerkungen zur Diracschen Theorie des Positrons. Z. Phys. 90, 209–231 (1934) 32. Hundertmark, D., R¨ohrl, N., Siedentop, H.: The sharp bound on the stability of the relativistic electron-positron field in Hartree-Fock approximation. Commun. Math. Phys. 211(3), 629–642 (2000) 33. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. New York: McGraw-Hill, 1980 34. Kim, Y.K.: Relativistic self-consistent Field theory for closed-shell atoms. Phys. Rev. 154, 17–39 (1967) 35. Klaus, M.: Non-regularity of the Coulomb potential in quantum electrodynamics. Helv. Phys. Acta 53, 36–39 (1980) 36. Klaus, M., Scharf, G.: The regular external field problem in quantum electrodynamics. Helv. Phys. Acta 50, 779–802 (1977) 37. Klaus, M., Scharf, G.: Vacuum polarization in fock space. Helv. Phys. Acta 50, 803–814 (1977) 38. Lieb, E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett. 46, 457–459 (1981) 39. Lieb, E.H., Siedentop, H.: Renormalization of the regularized relativistic electron-positron field. Commun. Math. Phys. 213(3), 673–683 (2000) 40. Lieb, E.H., Simon, B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977) 41. Lindgren, I., Rosen, A.: Relativistic self-consistent field calculations. Case Stud. At. Phys. 4, 93–149 (1974) 42. Mittleman, M.H.: Theory of Relativistic effects on atoms: Configuration-space Hamiltonian. Phys. Rev. A 24(3), 1167–1175 (1981) 43. Mohr, P.J., Plunien, G., Soff, G.: QED corrections in heavy atoms. Phys. Rep. 293, 227–369 (1998) 44. Paturel, E.: Solutions of the Dirac-Fock equations without projector. Ann. Henri Poincar´e 1(6), 1123–1157 (2000) 45. Pauli, W., Rose, M.E.: Remarks on the polarization effects in the positron theory. Phys. Rev II 49, 462–465 (1936) 46. Reinhardt, J., M¨uller, B., Greiner, W.: Theory of positron production in heavy-ion collision. Phys. Rev. A 24(1), 103–128 (1981)
562
´ S´er´e C. Hainzl, M. Lewin, E.
47. Ruijsenaars, S.N.M.: On Bogoliubov transformations for systems of relativistic charged particles. J. Math. Phys. 18(3), 517–526 (1977) 48. Simon, B.: Trace Ideals and their Applications. Vol. 35 of London Mathematical Society Lecture Notes Series. Cambridge: Cambridge University Press, 1979 49. Scharf, G., Seipp, H.P.: Charged vacuum, spontaneous positron production and all that. Phys. Lett. 108B(3), 196–198 (1982) 50. Seipp, H.P.: On the S-operator for the external field problem of QED. Helv. Phys. Acta 55, 1–28 (1982) 51. Serber, R.: Linear modifications in the Maxwell field equations. Phys. Rev., II. Ser. 48, 49–54 (1935) 52. Shale, D., Stinespring, W.F.: Spinor representations of infinite orthogonal groups. J. Math. Mech. 14, 315–322 (1965) 53. Swirles, B.: The relativistic self-consistent field. Proc. Roy. Soc. A 152, 625–649 (1935) 54. Thaller, B.: The Dirac Equation. Berlin-Heidelberg-New York: Springer Verlag, 1992 55. Uehling, E.A.: Polarization effects in the positron theory. Phys. Rev., II. Ser. 48, 55–63 (1935) ¨ 56. Weisskopf, V.: Uber die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons. Math.-Fys. Medd., Danske Vid. Selsk. 16(6), 1–39 (1936) 57. Wolkowisky, J.H.: Existence of solutions of the Hartree equations for N electrons. An application of the Schauder-Tychonoff theorem. Indiana Univ. Math. J. 22, 551–568 (1972-73) Communicated by H.-T. Yau
Commun. Math. Phys. 257, 563–578 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1350-5
Communications in
Mathematical Physics
Formality for Lie Algebroids Damien Calaque IRMA, 7 rue René Descartes, 67084 Strasbourg, France. E-mail: [email protected] Received: 29 April 2004 / Accepted: 20 December 2004 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: Using Dolgushev’s generalization of Fedosov’s method for deformation quantization, we give a positive answer to a question of P. Xu: can one prove a formality theorem for Lie algebroids ? As a direct application of this result, we obtain that any triangular Lie bialgebroid is quantizable.
Introduction The main goal of this paper is to formulate precisely a ‘Kontsevich-like’ (see [K]) formality theorem for Lie algebroids and then prove it. This problem has been proposed by Ping Xu at the end of [X] (Question 2). To do it, we use a slightly modified version of Dolgushev’s equivariant globalisation of Kontsevich’s formality [Do1] (in his paper Dolgushev generalises Fedosov’s geometric construction of star-products [Fv] to the case of a general manifold). We would like to mention that most of the proofs given in this paper are similar to those of [Do1] and apologize for this repetition. The paper is organized as follows. Section 1 is devoted to the presentation of our main results. We first recall some basic facts about differential geometry for Lie algebroids (see [CW, M] for details). Then we prove a Hochschild-Kostant-Rosenberg theorem for Lie algebroids and state our formality theorem. We finally explain why this result implies that any triangular Lie bialgebroid is quantizable (which has been proved in the case of regular ones by Ping Xu [X], see also [NT]). In Sect. 2, we construct resolutions of the desired DGLA using a torsion free Lie algebroid connexion. It is the more technical part of the paper. We end the proof of our main theorem in Sect. 3: after twisting a fiberwize quasiisomorphism (given by [K]) we use the resolutions of the previous section to contract it to the desired one. We also prove an equivariant version of our result.
564
D. Calaque
We recall in an appendix some facts about L∞ -algebras, Hopf algebroids, and Lie algebroid connections. Throughout the paper the Einstein convention for summation over repeated indices is assumed. 1. Main Results 1.1. Preliminaries: differential geometry for Lie algebroids. Definition 1.1 ([M]). A Lie algebroid is a vector bundle E over a manifold X equipped with a Lie bracket [, ]E on sections (X, E) and a bundle map ρ : E → T X called the anchor such that: 1. The induced map ρ : (X, E) → (X, T X) is a Lie algebra morphism. 2. For any f ∈ C ∞ (X), v, w ∈ (X, E), [v, f w]E = f [v, w]E + (ρ(v) · f )w
(Leibniz identity).
Basic objects in differential geometry are tensors. So it is natural to consider their algebroids analogues which we call E-tensors: for k, l ≥ 0, an E-(k, l)-tensor is a section of the bundle (⊗k E) ⊗ (⊗l E ∗ ). In a local base (e1 , . . . , er ) of E with dual base (ξ 1 , . . . , ξ r ) of E ∗ , such an E-tensor T can be written ...ik T (x) = Tji11...j (x)ei1 ⊗ · · · ⊗ eik ⊗ ξ j1 ⊗ · · · ⊗ ξ jl . l
Indices i1 , . . . , ik and j1 , . . . , jl are respectively called contravariant and covariant. As in usual differential geometry, one can consider the graded commutative algebra of E-differential forms E (X) := (X, ∧E ∗ ), which is endowed with a square zero super-derivation dE : E ∗ (M) → E ∗+1 (M). In local E-coordinates, any E-k-form ω can be written ω(x) = ωi1 ...ik (x)ξ i1 ∧ · · · ∧ ξ ik , where ωi1 ...ik are coefficients of a covariant E-tensor antisymmetric in indices i1 , . . . , ik , and 1 ∂ k (x) k , dE = ξ i ρ(ei ) − ξ i ∧ ξ j cij 2 ∂ξ k (x)e . where [ei , ej ]E = cij k In the same way, one can define the differential graded Lie algebra (DGLA for short) of E-polyvector fields k Tpoly E = Tpoly E= (X, ∧k+1 E) k≥−1
k≥−1
endowed with the zero differential and the Lie super-bracket of degree zero which extend k E, v ∈ T l uniquely [, ]E as follow: for u ∈ Tpoly poly E and w ∈ Tpoly E, [u, v ∧ w]E = [u, v]E ∧ w + (−1)k(l+1) v ∧ [u, w]E . As in the case of E-forms, any E-k-vector field v can be written locally v(x) = v i1 ...ik (x)ei1 ∧ · · · ∧ eik with v i1 ...ik coefficients of a contravariant E-tensor antisymmetric in indices i1 , . . . , ik .
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The usual algebra of differential operators can be viewed as a kind of universal enveloping algebra of the Lie algebra of vector fields. Let us define in a similar way the algebra of E-differential operators as the quotient of the graded algebra freely generated1 by C ∞ (X) (of degree 0) and (X, E) (of degree 1) by relations f ⊗ g − f g, f ⊗ v − f v, v ⊗ f − f ⊗ v − ρ(v) · f, v1 ⊗ v2 − v2 ⊗ v1 − [v1 , v2 ]E
f, g ∈ C ∞ (X), f ∈ C ∞ (X), v ∈ (X, E), f ∈ C ∞ (X), v ∈ (X, E), vi ∈ (X, E).
U E carries a natural Hopf algebroid structure (see Appendix B) with base algebra C ∞ (X), source and target maps s = t : C ∞ (X) → U E the natural inclusion, coproduct U E (where ⊗ denotes ⊗C ∞ (X) ) which extends : U E → U E⊗ 1 = 1⊗ f, ∀f ∈ C ∞ (X), (f ) = f ⊗ 1 + 1⊗ v, ∀v ∈ (X, E), (v) = v ⊗
(1)
and counit : U E → C ∞ (X) which extends (f ) = f, ∀f ∈ C ∞ (X) and (v) = 0, ∀v ∈ (X, E). This allows us to define a Lie super-bracket on the graded vector space k Dpoly E= U E ⊗k+1 Dpoly E = k≥−1
k≥−1
of E-polydifferential operators in a way similar to Appendix A.2: for homogeneous ki elements Pi ∈ Dpoly (X) (i = 1, 2), [P1 , P2 ] = P1 • P2 − (−1)k1 k2 P2 • P1 , where P1 • P2 =
k1
k2 +1 ⊗ id⊗k1 −i (P1 ) · (1⊗i ⊗ P2 ⊗ 1⊗k1 −i ). (−1)ik2 id⊗i ⊗
i=0
1 ∈ Dpoly E is such that [m0 , m0 ] = 0, thus (∂ = [m0 , ·], [, ]) Remark that m0 = 1⊗ defines a DGLA structure on Dpoly E. By an easy calculation, one can observe that ∂ is ∗ U E. simply the Hochschild coboundary operator (up to a sign) for the complex ⊗ 1.2. Formality for Lie algebroids. Let E → X be a Lie algebroid. First, in the spirit of the Hochschild-Kostant-Rosenberg theorem we are going to ∗ E (which was first proved in [V] for E = T X). prove that H ∗ (Dpoly E, ∂) ∼ = Tpoly Theorem 1.2. Define the map Uhkr : (Tpoly E, 0) → (Dpoly E, ∂) by Uhkr (v0 ∧ · · · ∧ vn ) =
1 (σ )vσ0 ⊗ · · · ⊗ vσn (n + 1)! σ ∈Sn+1
if n ≥ 0 and vi ∈ (X, E), and Uhkr (f ) = f if f ∈ C ∞ (X). It is a quasi-isomorphism of complexes (i.e., it is a morphism of complexes which induces an isomorphism in cohomology). 1 We consider here a completed tensor product: infinite sums which are finite on any compact are allowed.
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Proof. First, one can immediately check that the image of Uhkr is annihilated by ∂, i.e. that it is a morphism of complexes. Now remark that the complex Dpoly E is filtered by the total degree of polydifferential operators. Tpoly E carries also a natural filtration (which is in fact a gradation), namely by degree of polyvector fields. Then Uhkr is compatible with filtrations. Thus we have to prove that Gr(Uhkr ) : Gr(Tpoly E) → Gr(Dpoly E) is a quasi-isomorphism of complexes. In Gr(Dpoly E) all components are sections of some vector bundle on X and ∂ is C ∞ (X)-linear (the same is obviously true for Tpoly E), therefore we have to show that Gr(Uhkr ) is a quasi-isomorphism fiberwise. Fix x ∈ X and consider the vector space V = Ex . One has Gr(Dpoly E)x = S(V )⊗n , n≥0
but it is better to identify S(V ) with the cofree cocommutative coalgebra with counit C := C(V )⊕(R1)∗ . As above the differential can be expressed in terms of the cocommutative coproduct ; namely n−1
(−1)
∗
∂ = 1 ⊗ id
⊗n
−
n−1
(−1)i id ⊗ · · · ⊗ i ⊗ · · · ⊗ id + (−1)n−1 id⊗n ⊗ 1∗ .
i=1
Now let us recall a standard result in homological algebra: Lemma 1.3. Let C be the cofree cocommutative coalgebra with counit cogenerated by a vector space V . Then the natural homomorphism of complexes (∧∗ V , 0) → (⊗∗ C, ∂) is a quasi-isomorphism. Apply this lemma in the case when V = Ex and remark that Gr(Tpoly E)x = (Tpoly E)x = ∧∗ V . The theorem is proved. Now we claim that Dpoly E is formal: Theorem 1.4 (Formality). There exists a quasi-isomorphism UE of DGLA from (Tpoly E, 0, [, ]E ) to (Dpoly E, ∂, [, ]). When E = T X this is the formality theorem for manifolds presented in [K, Sect. 4.6]. More generally, if the anchor of E is injective then E ⊂ T X is an integrable distribution (i.e., X is foliated) and thus one obtains a leafwise formality for X. 1.3. Quantization of triangular Lie bialgebroids. Let E → X be a Lie algebroid with bracket [, ]E and anchor ρ. Let H = U E, R = C ∞ (X), defined by (1), s = t : R → U E be the natural embedding and ε : U E → R extending ε(f ) = f, ∀f ∈ R = C ∞ (X), ε(v) = 0, ∀v ∈ H = U E. It is a Hopf algebroid (see Appendix B). Moreover one can obviously extend the anchor to a map ρ : U E → U (T X) ⊂ End(R). It defines an anchor for the Hopf algebroid H . In [X] Ping Xu observes that any Hopf algebroid deformation of U E endows E with a Lie bialgebroid structure (by taking the semi-classical limit). Recall the
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Definition 1.5 ([MX]). A Lie bialgebroid is a Lie algebroid E → X whose dual bundle E ∗ → X is also a Lie algebroid and such that the differential dE ∗ on (X, ∧∗ E) is a derivation of the super-bracket [, ]E ; namely ∀v, w ∈ (X, E), dE ∗ [v, w]E = [dE ∗ v, w]E + [v, dE ∗ w]E . A Lie bialgebroid E is called triangular if dE ∗ = [, ·]E for a given ∈ (X, ∧2 E) satisfying [, ]E = 0. Reciprocally we say that a Lie bialgebroid is quantizable if there exists a deformation of U E whose semi-classical limit is precisely the starting bialgebroid structure. Conjecture 1.6. Any Lie bialgebroid is quantizable. Following [Dr], Xu shows in [X] that to quantize a triangular Lie bialgebroid it is suffiop cient to find a twistor (see Appendix B) J ∈ (U E ⊗R U E)[[]] such that J −J = mod and consider (U E[[]], RJ , J , sJ , tJ , ε). We construct such a J with the help of our formality Theorem 1.4. Theorem 1.7. Any triangular Lie bialgebroid is quantizable. Proof. Let us define J = m0 +
n n≥1
n!
UE[n] (, . . . , ).
1 Now since U is a L∞ -morphism α = (J − m0 ) ∈ (Dpoly E)[[]] is a Maurer-Cartan 1 element, ∂α + 2 [α, α] = 0. It means that
1 0 = [m0 , J ] − [m0 , m0 ] + ([J, J ] − [m0 , J ] − [J, m0 ] + [m0 , m0 ]) 2 1 1 = [m0 , J ] + ([J, J ] − 2[m0 , J ]) = [J, J ]. 2 2 Then remark that J 12,3 J 1,2 − J 1,23 J 2,3 = 21 [J, J ]. Finally, since U [1] is a quasi-isomorphism of complexes we have = Alt(U [1] ()) = J −J op + O(). 2. Dolgushev-Fedosov Resolutions of Tpoly E and Dpoly E Let E → X be a Lie algebroid with bracket [, ]E and anchor ρ. ˆ ∗ ), 2.1. The Weyl bundle and related bundles. Consider the bundle of algebras W = S(E whose sections are functions on E formal in the fibers. Any section s ∈ (X, W) can be written locally s = s(x, y) =
∞ l=0
si1 ...il (x)y i1 · · · y il ,
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where y i are formal coordinates on the fibers of E and si1 ...il are coefficients of a covariant E-tensor symmetric in indices i1 , . . . , il . k In the same way, one can define the bundle T = k≥−1 T of formal fiberwise k k+1 polyvector fields on E; T = W ⊗ ∧ E. Any homogeneous section v ∈ (X, T k ) is locally of the form v=
∞
j ...j
vi10...il k (x)y i1 · · · y il
l=0
∂ ∂ ∧ ··· ∧ j , j 0 ∂y ∂y k
j ...j
where vi10...il k are coefficients of an E-tensor symmetric in covariant indices i1 , . . . , il and antisymmetric in contravariant indices j0 , . . . , jk . Finally, we denote by D = k≥−1 Dk the bundle of formal fiberwise polydifferential operators on E; Dk = W ⊗ S(E)⊗k+1 . Any homogeneous section P ∈ (X, Dk ) is locally of the form P =
∞
...αk Piα1 0...i (x)y i1 · · · y il l
l=0
∂ |α0 | ∂ |αk | ⊗ · · · ⊗ , ∂y α0 ∂y αk
...αk where αs are multi-indices and Piα1 0...i are coefficients of an E-tensor symmetric in l covariant indices i1 , . . . , il . For our purposes, we need to tensor these bundles with the exterior algebra bundle ∧E ∗ . Namely, we need to consider the space E (X, B) of E-differential forms on X with values in B (from now, B will denote either W, T or D). In this setting, E (X, W) has a natural structure of super-commutative algebra, and E (X, T ) (resp. E (X, D)) is naturally endowed with the DGLA structure induced fiber-by-fiber by the DGLA structure of Tpoly (Rdf ormal ) (resp. Dpoly (Rdf ormal )). Let us denote the differential and the Lie super-bracket in E (X, D) by ∂ and [, ]G respectively, and the Lie super-bracket in E (X, T ) by [, ]S . In what follows we denote the same operations on these three different algebras by the same letters when it does not lead to any confusion. The differential δ = ξ i ∂y∂ i : E ∗ (X, W) → E ∗+1 (X, W) (δ 2 = 0) can obviously
extend to E (X, T ) and E (X, D). Namely, δ = [ξ i ∂y∂ i , ·]S on E (X, T ) and δ = [ξ i ∂y∂ i , ·]G on E (X, D). By definition, δ is a derivation of the Lie algebras E (X, T ) and E (X, D). Moreover, δ and ∂ super-commute since the multiplication operator m is δ-closed (δm = 0). Consequently, δ is compatible with DGLA structures on E (X, T ) and E (X, D). Proposition 2.1. For all n > 0, H n (E (X, B), δ) = 0, and H 0 (E (X, B), δ) = F 0 B is the space of sections of B that are constant in the fibers. Proof. Let us introduce the operator δ ∗ = y i ι(ei ) of contraction with the Euler vector field E = y i ei . On sections of E k (X, B) polynomial of degree l in the fibers, δδ ∗ + δ ∗ δ = (k + l)id (one can compute it in coordinates or use the Cartan formula for 1 ∗ the Lie derivative by E ). So we define the operator κ to be k+l δ on E-k-differential forms with value in B and l-polynomial in the fibers for k + l > 0, and 0 on sections of B constant in the fibers. Then one has u = δκu + κδu + Hu
u∈
E
(X, B),
(2)
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where Hu ∈ F 0 B is the harmonic part of u, that is to say its homogeneous part of zero exterior degree and constant in the fibers. 2.2. Flattening the connection. Let ∇ be a linear torsion free E-connection (it always exists). The connection defines a derivation of E (X, W) (which we will identify by the same symbol ∇). Denote by ijk (x) Christoffel’s symbols of ∇; thus one can write the induced derivation in local coordinates ∇ = dE + , where dE is as in Sect. 1.1 and = −ξ i ijk (x)y j ∂y∂ k . This derivation ∇ obviously extends to derivations of the DGLA Namely
E (X, D).
∇ = dE + [, ·]S : ∇ = dE + [, ·]G :
E
∗ (X, T ) → ∗ (X, D) →
E
E
E (X, T
) and
∗+1 (X, T ), ∗+1 (X, D).
E
On one hand it is clear by definition that ∇ is indeed a derivation of the Lie super-algebra structures. On the other hand dE (m) = 0 and [, m]G = 0 (this is just Leibniz rule), and hence ∇ super-commutes with ∂. k ), ∇ and δ super-commute: Since the connection is torsion free (i.e., ijk − jki = cij 1 k ∂ ∇δ + δ∇ = ξ i ∧ ξ j (ijk (x) − cij (x)) k = 0. 2 ∂y The standard curvature E-(1, 3)-tensor of the connection induces an operator R on which is given in local coordinates by
E (X, W)
∂ 1 R = − ξ i ∧ ξ j Rijl k (x)y k l : 2 ∂y
E
∗ (X, W) →
E
∗+2 (X, W),
where Rijl k are the coefficients of the curvature E-tensor (12). Then one has ∂ 1 m l m l l ∇ 2 = ξ i ∧ ξ j (ik j m + ρ(ej ) · ik + cij mk )y k l = R. 2 ∂y Obviously ∇ 2 acts as [R, ·]S and [R, ·]G respectively on E (X, T ) and E (X, D). Even though ∇ is not square zero in general, we use it to deform the differential δ. Namely, using an element A=
∞
j
ξ k Aki1 ...ip y i1 · · · y ip
p=2
∂ ∈ ∂y j
E
1 (X, T 0 ) ⊂
E
1 (X, D0 )
(3)
we construct a new derivation D = ∇ − δ + A : E ∗ (X, W) → E ∗+1 (X, W), D = ∇ − δ + [A, ·]S : E ∗ (X, T ) → E ∗+1 (X, T ), D = ∇ − δ + [A, ·]G : E ∗ (X, D) → E ∗+1 (X, D).
(4)
In some sense, ∇ can be viewed as a connection on the “big” bundles B which we flatten recursively by adding terms of higher polynomial degree in the fibers.
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Proposition 2.2. There exists an element A as in (3) such that κA = 0 and the corresponding derivation D (4) is square zero, D 2 = 0. In what follows, we write [A, ·] for A·, [A, ·]S , [A, ·]G when B is respectively W, T , D. Proof. Since κ raises the polynomial degree in the fibers (i.e., in y), there is a unique solution A in the form (3) to equation 1 A = κR + κ(∇A + [A, A]). 2
(5)
First observe that κ 2 = 0 implies that κA = 0. Now let us show that A satisfies equation 1 δA = R + ∇A + [A, A] 2
(6)
which obviously implies that D 2 = 0. Using (2) together with κA = 0 = HA one finds that 1 κδA = κR + κ(∇A + [A, A]). 2
(7)
Define C = −δA + R + ∇A + 21 [A, A]. Due to (7) κC = 0, and reformulating Bianchi’s identities for R one can show that δR = 0 = ∇R. These equalities, together with (2), imply that C = κ(∇C + [A, C]). Since the operator κ raises the polynomial degree in the fiber, this latter equation has a unique zero solution. Thus A satisfies (6) and the proposition is proved. 2.3. Acyclicity of the complexes. Resolutions. Theorem 2.3. H ∗ (E (X, B), D) = H 0 (E (X, B), D) ∼ = F 0 B. Proof. Using arguments similar to those of the proof of Proposition 2.2, one can show that H ∗ (E (X, B), D) = H 0 (E (X, B), D) (see [Do1, Theorem 3] for details). Let us now prove that H 0 (E (X, B), D) ∼ = F 0 B. For any u0 ∈ F 0 B, there is a unique solution u ∈ E 0 (X, B) = (X, B) of the equation u = u0 + κ(∇u + [A, u])
(8)
(still because κ raises the polynomial degree in the fibers). It is obvious that Hu = u0 ; let us prove that Du = 0. Let v = Du, then Dv = 0 = Hv, and κv = 0 after (8). Then use (2) to find v = κ(∇v + [A, v]). Again, this equation has a unique zero solution and consequently v = Du = 0. We have defined a linear map ϑ : F 0 B → Z 0 (E (X, B), D) = H 0 (E (X, B), D) that sends u0 to the solution u = ϑ(u0 ) of (8) and such that H(ϑ(u0 )) = u0 . This map is obviously injective: the solution of (8) is zero if and only if u0 = 0. It is also surjective: if v ∈ Z 0 (E (X, B), D) is such that Hv = 0 then by (2) v = κ(∇v + [A, v]) and so v = 0.
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In the case of the Weyl bundle B = W, one can easily show that H : H ∗ (E (X, W), D) = Z 0 (E (X, W), D) → F 0 W = C ∞ (X) is a morphism of commutative algebras. In the same spirit we have F 0 T = (X, ∧E) = Tpoly E. On the other hand the differential D respects the DGLA structure on E (X, T ) and thus its homology acquires a DGLA structure. In the following proposition we show that H respects the DGLA structures. Proposition 2.4. H ∗ (E (X, T ), D)) ∼ =DGLA Tpoly E. Proof. Let u, v ∈ Z 0 (E (X, T ), D). We are going to show that H([u, v]S ) = [H(u), H(v)]E .
(9)
Since H preserves the exterior product of polyvector fields it is sufficient to prove it in the following two cases: first when u0 = H(u) and v0 = H(v) are vector fields, next when u0 = H(u) a vector field and f = H(v) is a function. First case. Let u0 = ui (x) ∂y∂ i and v0 = v i (x) ∂y∂ i be vector fields. Note that k [u0 , v0 ]E = (ui ρ(ei )v k + ui v j cij − v i ρ(ei )uk )
∂ . ∂y k
Then, by an easy calculation one obtains u = ϑ(u0 ) = u + y i (ρ(ei )uk + ijk uj ) ∂y∂ k mod |y|2 , v = ϑ(v0 ) = v + y i (ρ(ei )v k + ijk v j ) ∂y∂ k mod |y|2 .
And thus [u, v]S = ui (ρ(ei )v k + ijk v j ) ∂y∂ k − v i (ρ(ei )uk + ijk uj ) ∂y∂ k mod |y| k ui v j − v i ρ(e )uk ) ∂ = (ui ρ(ei )v k + cij i ∂y k = ϑ([u0 , v0 ]E )
mod |y| mod |y|.
Second case. Let u0 = ui (x) ∂y∂ i be a vector field and f be a function. One has [u0 , f ]E = ρ(u0 )f = ui ρ(ei )f. Since v = ϑ(f ) = f + y i ρ(ei )f mod |y|2 we obtain [u, v]S = ui ρ(ei )f mod |y|. Consequently (9) is satisfied and H is an isomorphism of graded Lie algebras. Since the differentials are both zero it is a DGLA-isomorphism. As above D preserves the DGLA structure on E (X, D) and thus its homology is also a DGLA. Using the PBW theorem for Lie algebroids (see [R, NWX]) one finds that ∗ U (E) are isomorphic as (filtered) vector F 0 D = (X, ⊗∗ S(E)) and Dpoly E = ⊗ spaces. Again we have: Proposition 2.5. H ∗ (E (X, D), D) ∼ =DGLA Dpoly E.
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k Proof. Let us first set τ0 = 1 and τk+1 = y i ei τk − y i y j ijl (x) ∂τ . For any k ∈ N τk is a ∂y l ∞ well-defined element of (X, W) ⊗C (X) U E. Then for any fiberwise differential oper1 (H⊗id)(u·τk ) ator constant in the fibers u = ui1 ...ik (x) ∂y∂i1 · · · ∂y∂ik we define µ(u) := k! and compute 1 µ(u) = ui1 ...ik (x) eiσ1 · · · eiσk mod Uk−1 (E). k!
σ ∈Sk
Thus µ is filtered and its associated graded map coincides with the usual isomorphism (X, S k (E))→U ˜ k (E)/Uk−1 (E) (see [NWX, proof of Theorem 3]). Consequently µ is an isomorphism which naturally extends to an isomorphism from F 0 D to Dpoly E. Moreover, µ ◦ = ◦ µ on (X, S(E)). Next we also have H ◦ = ◦ H on Z 0 (E (X, D0 ), D) and thus the composition µ ◦ H : Z 0 (E (X, D), D) → Dpoly E is compatible with coproducts. In particular it commutes with differentials: (µ ◦ H) ◦ ∂ = ∂ ◦ (µ ◦ H). Finally, due to the compatibility with coproducts and to the special form of the brackets (see 1.1 and A.2) it is now sufficient to show that µ ◦ H restricts to a morphism of associative algebras between Z 0 (E (X, D0 ), D) and U (E); we have to prove that 1 P i = ϑ ◦ µ−1 (Pi ). There 2 ), where Pi are generators of U (E) and P P1 P2 = µ ◦ H(P are four distinct cases: First case. P1 = ui (x)ei and P2 = v j (x)ej are vector fields. Then µ−1 (P1 ) = ui ∂y∂ i , 1 P 2 ) = ui v j ∂i j + ui (ρ(ei )v k + k v j ) ∂ k . And µ−1 (P2 ) = v j ∂y∂ j , and thus H(P ij ∂y ∂y ∂y 2
2
since µ(ui v j ∂y∂i ∂y j ) = 21 ui v j (ei ej + ej ei − (ijk + jki )ek ) one computes 1 i j k (u v (ei ej + ej ei + cij ek )) + ui ρ(ei )v j ej 2 ∂2 ∂ 2 ). 1 P = µ(ui v j i j + ui (v j ijk + ρ(ei )v k ) k ) = µ ◦ H(P ∂y ∂y ∂y
P1 P2 = ui ei v j ej =
Second and third cases. P1 = ui (x)ei is a vector field and P2 = f is a function. Since µ−1 (P1 ) = f we have P1 P2 = ui (f ei + ρ(ei )f ) = µ(ui (f ∂y∂ i + ρ(ei )f )) = 2 P 2 ) and P2 P1 = f ui ei = µ(f ui ∂ i ) = µ ◦ H(P 1 ). 1 P µ ◦ H(P ∂y
1 P 2 ). Fourth case. P1 = f and P2 = g are functions. P1 P2 = f g = µ ◦ H(P Consequently µ ◦ H is a DGLA-isomorphism. 3. Proof of Theorem 1.4 3.1. Twisting a fiberwise quasi-isomorphism. In virtue of Properties 1 and 2 in Theorem A.3 we have a fiberwise quasi-isomorphism UK from (E (X, T ), 0, [, ]S ) to (E (X, D), ∂, [, ]G ). Our purpose is to twist UK in order to get a quasi-isomorphism from (E (X, T ), D, [, ]S ) to (E (X, T ), ∂ + D, [, ]G ). U Let us recall that the differential D can be written locally in the form D = dE + [B, ·]S : E ∗ (X, T ) → E ∗+1 (X, T ) D = dE + [B, ·]G : E ∗ (X, D) → E ∗+1 (X, D) where B = −ξ i ∂y∂ i − ξ i ijk (x)y j ∂y∂ k + p≥2 ξ i Akij1 ...jp (x)y j1 · · · y jp ∂y∂ k
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Let V be a E-coordinates disk, then we prove Proposition 3.1. UK defines a quasi-isomorphism of DGLA from (E (V , T ), dE , [, ]S ) to (E (V , D), ∂ + dE , [, ]G ). Proof. Let us note respectively T and D for E (V , T ) and E (V , D). Since dE commutes with the fiberwise DGLA structures of T and D, and also with the fiberwise L∞ morphism UK , then UK defines a L∞ -morphism from (T, dE , [, ]S ) to (D, ∂ +dE , [, ]G ). Now observe that (T, 0, dE ) and (D, ∂, dE ) are double complexes; UK[1] = Ahkr is an inclusion of double complexes. Thus we have a long exact sequence in cohomology · · · → H k (T, dE ) → H k (D, ∂ + dE ) → H k (D/T, ∂ + dE ) → · · · . Since the inclusion Ahkr : (T, 0) → (D, ∂) is a quasi-isomorphism of complexes one has H ∗ (D/T, ∂) = 0. Then the (second) spectral sequence of the double complex (D/T, ∂, dE ) goes to zero and thus H ∗ (D/T, ∂ + dE ) = 0. Consequently Ahkr induces an isomorphism H ∗ (T, dE )→H ˜ ∗ (D, ∂ + dE ). It means that UK is a quasi-isomorphism of DGLA from (T, dE , [, ]S ) to (D, ∂ + dE , [, ]G ). On V the element B ∈E (V , T 0 ) ⊂E (V , D0 ) is well-defined, and since D is square zero it is a Maurer-Cartan element. It means that (E (V , T ), D, [, ]S ) (resp. (E (V , D), ∂ + D, [, ]G )) can be obtained using a twisting of the DGLA (E (V , T ), dE , [, ]S ) (resp. (E (V , D), ∂ + dE , [, ]G )) by B (a general description of twisting procedures for DGLA and their L∞ -morphisms is presented in [Do2, Sect. 2.3]). Due to Properties 3 and 5 in TheoremA.3, UK maps B ∈E (V , T 0 ) to B ∈E (V , D0 ). of DGLA from (E (V , T ), D, [, ]S ) to Then we can define a quasi-isomorphism U E ( (V , T ), ∂ + D, [, ]G ) using a twisting of UK by B. Namely, (Y ) = exp((−B)∧)UK (exp(B∧)Y ). U
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does not depend on the choice of local coordinates and Next proposition tells us that U thus is defined globally. extends to a quasi-isomorphism of DGLA from (E (X, T ), D, [, ]S ) Proposition 3.2. U E to ( (X, D), ∂ + D, [, ]G ). Proof. See [Do1, Prop. 3] (it makes use of Property 4 in Theorem A.3).
. On one hand we know from Proposition 2.4 that 3.2. End of the proof: contraction of U there exists a quasi-isomorphism of DGLA UT from (Tpoly E, 0, [, ]E ) to (E (X, T ), D, [, ]S ). On the other hand we have also a quasi-isomorphism of DGLA from (E (X, T ), D, [, ]S ) to (E (X, D), ∂ + D, [, ]G ) (Sect. 3.1). Let us define U ◦ UT and claim U =U Proposition 3.3. One can modify U to construct a quasi-isomorphism of DGLA U from Tpoly E to E (X, D) whose structure maps take values in Z 0 (E (X, D), D). Proof. See [Do1, Prop. 5].
Consequently, composing U with the DGLA-isomorphism of Proposition 2.5 we obtain a quasi-isomorphism of DGLA UE from (Tpoly E, 0, [, ]E ) to (Dpoly E, ∂, [, ]). Thus we have proved Theorem 1.4. Q.E.D.
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3.3. Equivariant formality theorem. By a good action of a group G on a Lie algebroid E → X we mean a smooth action, linear in the fibers and compatible with anchor map and bracket: for all g ∈ G, u, v ∈ E, g · ρ(u) = ρ(g · u) and g · [u, v]E = [g · u, g · v]E . Such an action extends naturally to Tpoly E and Dpoly E with the property that all structures are G-invariant. In this context the quasi-isomorphism of complexes defined in Theorem 1.2 is G-equivariant. In particular it restricts to a quasi-isomorphism of complexes Uhkr : (Tpoly E)G → (Dpoly E)G . The following theorem is a G-equivariant version of Theorem 1.4. Theorem 3.4. Consider a Lie algebroid E → X equipped with a good action of a group G. If there exists a G-invariant torsion free E-connexion ∇, then one can construct a G-equivariant quasi-isomorphism of DGLA from Tpoly E to Dpoly E. Proof. First one can canonically extend the action of G to the spaces E (M, W), ) and E (M, D) in such a way that all algebraic structures we have defined are G-invariant. First we are going to prove that the resolutions constructed in Sect. 2.3 are G-equivariant. The differential δ, the homotopy operator κ and the projection H are obviously G-invariant. The G-invariance of the connection ∇ implies the G-invariance of the induced derivation (also called ∇) and of the curvature tensor R. Thus Eq. (5) has a G-invariant solution A (3) and then the differential D (4) of Proposition 2.2 is G-invariant. In the same way, ϑ is G-invariant since it is defined by G-equivariant Eq. (8). Thus the DGLA-isomorphisms of Propositions 2.4 and 2.5 are G-equivariant. Second, since G acts on the fibers by linear transformations and due to Property 2 in Theorem A.3 the fiberwise quasi-isomorphism UK is G-equivariant. constructed with the help of Third we have to prove that the quasi-isomorphim U the twisting procedure (10) is G-equivariant. Let V be a coordinate disk and B be the twisting element of Sect. 3.1. Since UK is G-equivariant one has E (M, T
gUK[n+m] (B, . . . , B, v1 , . . . , vn ) = UK[n+m] (gB, . . . , gB, gv1 , . . . , gvn ), to be where g ∈ G acts on B as on a tensor element. Now a sufficient condition for U G-equivariant is gUK[n+m] (B, . . . , B, v1 , . . . , vn ) = UK[n+m] (g · B, . . . , g · B, gv1 , . . . , gvn ), where g· acts by ususal transformations of Christoffel’s symbols in B. Then remark that g · B − gB is a fiberwise polyvector field linear in the fibers on V ; thus using Property 4 of Theorem A.3 we obtain the desired result. Finally, it is not difficult to see that the contraction procedure of Sect. 3.2 involves only G-equivariant cohomological equations (see [Do1] for details). Example. 1) Consider the case of a Lie algebra g (i.e., a Lie algebroid over a point) with the adjoint action of its Lie group G (which is a good action). Then the Lie algebroid connection given by half the Lie bracket on g is a torsion free G-invariant connection and we obtain a G-equivariant quasi-isomorphism of DGLA from ∧∗ g to ⊗∗ U g. In particular for any subgroup H ⊂ G one obtains a quasi-isomorphism of DGLA from (∧∗ g)H to (⊗∗ U g)H . 2) If a group G acts smoothly on a manifold X, then it induces a good action on T X. In this particular case our theorem is equivalent to Theorem 5 of [Do1].
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3) Now if E → X is a Lie algebroid with injective anchor (i.e., E is the Lie algebroid of a foliation), then any smooth action of a group G on X that respects the foliation (i.e., that sends a leaf to a leaf) gives rise to a good action on E. In this context we obtain a leafwise version of the previous example. Acknowledgements. I am grateful to my advisor, B. Enriquez, who has accepted to lead my research and read carefully this paper. I am also greatly indebted to G. Halbout for teaching me the ideas of [Do1, Fv, K]. Discussions on ‘thing-oids’ with P. Xu in Normandie were very enlightening, I express to him my sincere thanks. I also thank V. Dolgushev for his warned comments.
A. Formality, L∞ and All That A.1. Quasi-isomorphisms of differential graded Lie algebras. Let (g, d, [, ]) be a differential graded Lie algebra (DGLA). We assume that the differential is of degree one and the Lie super-bracket is of degree zero. One can associate to g a cocommutative coalgebra C∗ (g[1]) cofreely generated by the vector space g with a shifted parity, equipped with a coderivation Q having two non-vanishing structure maps Q[1] = d : g → g[1] and Q[2] = [, ] : ∧2 g → g. The fact that (g, d, [, ]) is a DGLA is equivalent to the nilpotency of Q (i.e., Q2 = 0). Definition A.1. A L∞ -morphism between two DGLA (g1 , d1 , [, ]1 ) and (g2 , d2 , [, ]2 ) is a morphism of cocommutative coalgebras L : C∗ (g1 ) → C∗ (g2 ) compatible with the DGLA structures in the following sense: Q2 ◦ L = L ◦ Q1 , where Qi is the square zero coderivation corresponding to (di , [, ]i ). Definition A.2. A quasi-isomorphism of DGLA from (g1 , d1 , [, ]1 ) to (g2 , d2 , [, ]2 ) is a L∞ -morphism U from g1 to g2 whose first structure map U [1] : g1 → g2 induces an isomorphism in cohomology H ∗ (g1 , d1 ) ∼ = H ∗ (g2 , d2 ). A DGLA is formal if it is quasi-isomorphic to the graded Lie algebra (with zero differential) of its cohomology. A.2. Kontsevich formality theorem. Let Dpoly (X) be the vector space of polydifferential operators on a smooth manifold X. It is a graded vector space k Dpoly (X), Dpoly (X) = k≥−1 k where Dpoly (X) denotes the subspace of operators of rank k +1. We define on Dpoly (X) a Lie super-bracket (the Gerstenhaber bracket) given on homogeneous elements by ki Pi ∈ Dpoly (X) (i = 1, 2) by [P1 , P2 ]G = P1 • P2 − (−1)k1 k2 P2 • P1 , where
P1 • P2 (f0 , . . . , fk1 +k2 )=
k1
(−1)ik2 P1 (f0 , . . . , fi−1 , P2 (fi , . . . , fi+k2 ), . . . , fk1 +k2 ).
i=0 1 (X) can be written in Associativity condition for the multiplication operator m0 ∈ Dpoly terms of the Gerstenhaber bracket as [m0 , m0 ]G = 0. Thus (∂ = [m0 , ·]G , [, ]G ) defines a DGLA structure on Dpoly (X).
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Let now Tpoly (X) be the DGLA of polyvector fields on X: k Tpoly (X) = (X, ∧k+1 T X) Tpoly (X) = k≥−1
k≥−1
endowed with the standard Schouten-Nijenhuis bracket and zero differential. Hochschild-Kostant-Rosenberg theorem says that the antisymmetrisation map Ahkr : ∗ (X), and Tpoly (X) → Dpoly (X) induces an isomorphism H ∗ (Dpoly (X), ∂) ∼ = Tpoly Kontsevich has proved in [K] that Dpoly (X) is formal. We will use a version of this result when X = Rdf ormal : Theorem A.3 (Kontsevich,[K]). There exists a quasi-isomorphism of DGLA UK from Tpoly (Rd ) to Dpoly (Rd ) which has the following properties: 1. UK can be defined for Rdf ormal (the formal completion of Rd at the origin) as well. 2. UK is GLd (R)-equivariant. [n] 0 (Rd 3. For any n ≥ 2, v1 , . . . , vn ∈ Tpoly f ormal ), UK (v1 , . . . , vn ) = 0. d 0 (Rd 4. For any n ≥ 2, v ∈ gld (R) ⊂ Tpoly f ormal ), χ2 , . . . , χn ∈ Tpoly (Rf ormal ), UK[n] (v, χ2 , . . . , χn ) = 0. 5. UK[1] = Ahkr . B. Hopf Algebroids Definition B.1 ([X], see also [L]). A Hopf algebroid is an associative algebra with unit H together with a base algebra R, an algebra homomorphism s : R → H and an algebra antihomomorphism t : R → H whose respective images commute together (the source and target maps, which give H an R-bimodule structure), and R-bimodule maps : H → H ⊗R H (the coproduct) and ε : H → R (the counit) such that 1. (1) = 1 ⊗R 1 and ( ⊗R id) ◦ = (id ⊗R ) ◦ , 2. ∀a ∈ R, ∀h ∈ H, (h)(t (a) ⊗R 1 − 1 ⊗R s(a)) = 0, 3. ∀h1 , h2 ∈ H, (h1 h2 ) = (h1 )(h2 ), 4. ε(1H ) = 1R and (ε ⊗R idH ) ◦ = (idH ⊗R ε) ◦ = idH . Given a Hopf algebroid H over a base R, an anchor is a representation ρ : H → End(R) which is also a R-bimodule map and satisfies s(ρ(x1 ) · a)x2 = xs(a) x ∈ H, a ∈ R, x1 t (ρ(x2 ) · a) = xt (a) x ∈ H, a ∈ R, ρ(x) · 1R = ε(x) x ∈ H. A twistor ([X]) in a Hopf algebroid H over a base R is an invertible element J ∈ H ⊗R H that satisfies J 12,3 J 1,2 = J 1,23 J 2,3 , (ε ⊗R id)(J ) = (id ⊗R ε)(J ) = 1H .
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Let H be a Hopf algebroid over a base R (resp. with anchor ρ), and let J = i xi ⊗R yi be a twistor. Then one can define a new product on R given by a∗J b = i (ρ(xi )a)(ρ(yi )b), anew coproduct J = J −1 J , and new source and target maps given by sJ (a) = i s(ρ(xi )a)yi and tJ (a) = t (ρ(yi )a)xi . Denote RJ = (R, ∗J ).
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Theorem B.2 ([X], Theorem 4.14). Let (H, R, , s, t, ε) be a Hopf algebroid (resp. with anchor ρ). If J is a twistor, then (H, RJ , J , sJ , tJ , ε) is again a Hopf algebroid (resp. with the same anchor ρ).
C. Lie Algebroid Connections Let (E, [, ]E , ρ) be a Lie algebroid over a smooth manifold X. Definition C.1. A linear E-connection is a map ∇ : (X, E) × (X, E) → (X, E) such that 1. ∇ is C ∞ (X)-linear with respect to the first argument. 2. ∇ is R-linear with respect to the second argument. 3. for all f ∈ C ∞ (X) and u, v ∈ (X, E), ∇u f v = f ∇u v + (ρ(u) · f )v. In a local base (e1 , . . . , er ) of E, ∇ is completely determined by its Christoffel’s symbols ijk which are given by: ∇ei ej = ijk ek . Remark. As with usual connections, one can define the covariant derivative on E-tensor in a unique way such that ∇u is a derivation with respect to the tensor product of E-tensors, commutes with the contraction of E-tensors, acts as ρ(u) on functions, and is R-linear. Definition C.2. 1. The torsion T of ∇ is the E-(1, 2)-tensor defined by T (u, v) = ∇u v − ∇v u − [u, v]E . 2. The curvature R of ∇ is the E-(1, 3)-tensor defined by R(u, v)w = ([∇u , ∇v ] − ∇[u,v]E )w. Coefficients of these tensors can be expressed in a local base (e1 , . . . , en ): k, Tijk = ijk − jki − cij l l m m l − cm l . Rij k = im j k − ik jl m + ρ(ei ) · jl k − ρ(ej ) · ik ij mk
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Proposition C.3. There exists a torsion free linear E-connection. Proof. Let (Uα )α be a covering of X by trivializing opens for E. On each Uα one has (α) a basis (ei )i of sections and then can define ∇ei ej = 21 [ei , ej ]. Let fα be such that (α) . ∇ is a torsion free linear E-connection. α fα = 1 and define ∇ = fα ∇ Proposition C.4 (Bianchi’s identities). For all u, v, w ∈ (X, E) , ∇u R(v, w) + R(T (u, v), w) + c.p.(u, v, w) = 0,
R(u, v)w − T (T (u, v), w) − ∇u T (v, w) + c.p.(u, v, w) = 0. Proof. See for example [Fs].
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References [CW]
Cannas da Silva, A., Weinstein, A.: Geometric models for noncommutative algebras. Berkeley Mathematics Lecture Notes, Providence, RI: Amer. Math. Soc. 1999 [Do1] Dolgushev, V.: Covariant and equivariant formality theorems. Adv. Math. 191(1), 147–177 (2005) [Do2] Dolgushev, V.: A formality theorem for chains. http://arxiv.org/list/math.QA/0402248, 2004 to appear in Adv. Math. [Dr] Drinfeld, V.G.: On some unsolved problems in quantum group theory. Lect. Notes Math. 1510, 1–8 (1992) [Fv] Fedosov, B.: A simple geometric construction of deformation quantization. J. Diff. Geom. 40, 213–238 (1994) [Fs] Fernandes, R.L.: Lie algebroids, holonomy and characteristic classes. Adv. Math. 170, 119–179 (2002) [K] Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157– 216 (2003) [L] Lu, J.-H.: Hopf algebroids and quantum groupoids. Internat. J. Math. 7, 47–70 (1996) [M] Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry. London Math. Soc. Lecture Notes Series 124, Cambridge: Cambridge Univ. Press, 1987 [MX] Mackenzie, K., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73, 415–452 (1994) [NT] Nest, R., Tsygan, B.: Formal deformations of symplectic Lie algebroids, deformations of holomorphic structures and index theorems. Asian J. of Math. 5(4), 599–633 (2001) [NWX] Nistor, V., Weinstein, A., Xu, P.: Pseudodifferential operators on differential groupoids. Pacific J. Math. 189, 117–152 (1999) [R] Rinehart, G.S.: Differential forms on general commutative algebras. Trans. Amer. Math. Soc. 108, 195–222 (1963) [V] Vey, J.: Déformation du crochet de Poisson sur une variété symplectique. Comment Math. Helv. 50, 421–454 (1975) [X] Xu, P.: Quantum groupoids. Comm. Math. Phys. 216, 539–581 (2001) Communicated by L. Takhtajan
Commun. Math. Phys. 257, 579–619 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1351-4
Communications in
Mathematical Physics
The Green’s Function of the Navier-Stokes Equations for Gas Dynamics in R3 David Linnan Li Department of Mathematics, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected] Received: 16 May 2004 / Accepted: 31 January 2005 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: In this paper we derive pointwise estimates for the Green’s function of the Navier-Stokes equations for the compressible fluid. Our analysis shows that the short time behavior of the Green’s function is dominated by the high frequency waves but the large time behavior is dictated by low frequency waves. Furthermore, the low frequency waves consist of entropy and acoustic waves that demonstrate a weaker form of Huygens’ Principle. 1. Introduction Consider the Navier-Stokes equations for the compressible fluid: ρt + div(ρu) = 0, (ρu)t + div(ρuuT ) + ∇x P = div T , (ρE) + div(ρEu + P u) = div(uT ) + div(k(θ )∇ θ), t x
(1.1)
where the stress tensor T is defined as follows T = (∇((θ )u) + (∇((θ )u))T ) + div ((η(θ ) − (θ ))uI ). Here ρ(x, t), u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t))T and E(x, t) represent the fluid density, velocity and total energy, respectively. Thus m = ρu is the momentum density and = ρE is the energy density. Let e = E − |u|2 /2 be the internal energy. For simplicity, we consider the monatomic gas model, i.e., the pressure P = 23 ρe and the 2e temperature θ = 3R , where (θ ) > 0 and η(θ ) > 0 are viscosity coefficients, k(θ ) is the heat conductivity, and R is the ideal gas constant. We now linearize (1.1) about the constant state, which without loss of generality is taken to be z∗ = (1, 0, 1)T , ρt + div m = 0, (1.2) mt + 23 ∇w = m + η∇div m, + 5 div m = ζ ρ + κ, t 3
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where = (θ0 ), η = η(θ0 ), ζ = k(θ0 )θρ (1, 0, 1) < 0, κ = k(θ0 )θ (1, 0, 1) = −ζ > 0 and θ0 = θ (z∗ ). Set z = (ρ, m, )T . We can write Eq. (1.2) as (1.2 )
zt = Az,
where A is the differential operator corresponding to the linearized Navier-Stokes equations (1.2). The Green’s function G(x, t) is a 5 × 5 matrix that satisfies Gt = AG, (1.3) G(x, 0) = δ(x)I, where δ(x) is the Dirac delta function and I is the 5 × 5 identity matrix. Our goal is to derive pointwise estimates for G(x, t). In this paper, we decompose the Green’s function into low and high frequency waves in the Fourier space. Our analysis shows that the large time behavior of G(x, t) is dominated by the low frequency waves while the high frequency waves play a much more significant role in short time. Furthermore, we showed that the low frequency waves consist of entropy and acoustic waves. The acoustic waves travel at the sound speed c while the entropy waves permeate inside the wave cone |x| < c. Due to the hyperbolic nature of the Navier-Stokes equations, the Green’s function G(x, t) also contains Dirac delta functions which decay exponentially in time. To properly define these singularities, we shall now introduce the notion of pseudo-differential operators. Let f (x) be a distribution and fˆ(ξ ) its Fourier transform. Then the pseudo-differential operator χR (D) is defined as follows: χR (D)f (x) = eix·ξ fˆ(ξ )dξ. (1.4) |ξ |≥R
Notice that, in the above inversion formula, the integral is only taken on |ξ | ≥ R. Therefore, χR (D)f (x) represents only the high frequency waves. We shall now give the main result of this paper in the following theorem. Theorem 1.1 (Main Theorem). Let G(x, t) be the Green’s function for the compressible Navier-Stokes equations (1.1). For any multi-index α, there exist a distribution Fα and a constant wave speed c > 0 such that for sufficiently large R and for all x ∈ R3 , t ≥ 0, we have α |α|+3 √ D (G(x, t) − χR (D)Fα (x, t)) ≤ Cα,N t − 2 B 3 (|x|, t)χ x {|x|≤ct+ t} + BN (|x|, t) 2 1 + (1 + t)− 2 BN (|x| − ct, t) , (1.5) where
−N |x|2 BN (|x|, t) = 1 + . t
(1.6)
|α|+2
Here, N can be arbitrarily large and Fα has the form Fα (x, t) =
j =0
where
Lj δ(x) e−Ct ,
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1 00 = 0 0 0, L0 (t, D) 5/3 0 0 T 0 0 p11,2 (t)D = −−1 p2,1 (t)D L1 (t, D) 0 0, 1 3,2 T 0 p1 (t)D 0 1,1 p1 (t) 0 p11,3 (t) = −−1 0 −p2,2 (t)−1 D D T L2 (t, D) 0 . 2 3,1 3,3 p1 (t) 0 p1 (t) For j ≥ 2,
(1.7)
(1.8)
(1.9)
= (−1)j −j L2j −1 (t, D)
and
T 0 pj1,2 (t)D 0 2,1 0 pj2,3 (t)D pj (t)D 3,2 T 0 0 pj (t)D
(1.10)
= (−1)j −j L2j (t, D)
pj1,1 (t) 0 pj1,3 (t) 2,2 D T 0 . 0 −pj +1 (t)−1 D 0 pj3,3 (t) pj3,1 (t)
(1.11)
= (Dx1 , Dx2 , Dx3 )T , −1 is the inverse Laplacian operator and In Eqs. (1.7)–(1.11), D ik pj (t)’s are polynomials in t of degree no larger than j . The kernel BN (|x|, t) is first introduced by Liu and Wang [5] and is used to characterize the dissipation of the diffusion waves in our analysis. √ Three types of waves are identified in Eq. (1.5). Inside the cone C = {(x, t) : |x| < ct + t}, G(x, t) is dominated by entropy waves t −3/2 B3/2 (|x|, t) while outside the cone it consists of both entropy and acoustic waves: t −3/2 BN (|x|, t) for arbitrarily large N and t −3/2 (1 + t)−1/2 BN (|x| − ct, t). This demonstrates a weaker form of Huygens’Principle since G(x, t) decays more slowly on the wavefront |x| = ct than elsewhere. See Fig. 1 for illustration. We can also interpret Theorem 1.1 as Huygens’ Principle by examining the energy dissipation. From (1.5), we have 3
χ{|x|<√t} (G − χR (D)Fα ) L2 (x) ∼ t − 2 , 3
χ{||x|−ct|<√t} (G − χR (D)Fα ) L2 (x) ∼ t − 2 . We can also show that G − χR (D)Fα L2 (x) decays faster elsewhere, which indicates √ that energy propagates √ mainly on a “blurred” wavefront ||x| − ct| < t and inside a smaller ball |x| < t. This is also illustrated in Fig. 1. The nonlinear case will be considered in the future. In general, the solution to a nonlinear system may not be dictated by the Green’s function of the corresponding linear system. Liu and Zeng [6] showed that the large time behavior of the one dimensional Navier-Stokes equations is dominated by that of the Burgers’ equation. For multidimensional Navier-Stokes equations, however, we expect the behavior of the solution to be similar to that of the Green’s function, due to the presence of stronger dispersions in
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decay faster
decay rate t −3 |x| = at, a < c |x| = ct
√ t
√
t
|x| = ct
decay faster decay rate
Wave propagation
Energy dissipation
Fig. 1. Wave propagation and energy dissipation of the Green’s function
multidimensional spaces. Furthermore, the explicit and detailed structure of the pointwise estimates we obtained in (1.5) establishes the foundation and provides an important tool to study the nonlinear solutions. Large time behavior of the Navier-Stokes equations has been a central topic in studying compressible flows in gas dynamics. Kawashima [4] studied the asymptotic behavior of several general hyperbolic-parabolic systems and obtained L2 estimates. These estimates, while revealing the dissipative properties of the solutions, provide no information on wave propagation. To characterize the wave aspect of the solutions, one must resort to pointwise estimates. In the case of one space dimension, such estimates were obtained by Zeng [7] for the isentropic Navier-Stokes equations, and Liu and Zeng [6] for general quasilinear hyperbolic-parabolic systems of conservation laws. For multi-dimensional diffusion waves, Hoff and Zumbrun [3] studied the Green’s function of an artificial viscosity system associated with the isentropic Navier-Stokes equations and derived pointwise estimates. The Green’s function for isentropic Navier-Stokes equations has been studied by Liu and Wang [5]. In our analysis, the inclusion of the energy equation gives rise to much richer wave behavior. For instance, the interaction between the entropy and the acoustic waves is much more sophisticated. In [5], Liu and Wang provided a canonical way of identifying and estimating the two different types of waves. While this method still helps formally identify these waves in the compressible flow, their estimates need to be treated more carefully. In fact, when estimated separately, both waves contain singularities that would contradict our physical intuition. These singularities, however, are mere mathematical artifacts and would disappear when entropy and acoustic waves are combined. We shall now give an outline of the paper. In Sect. 2, we compute the Fourier transˆ ˆ back to the physical space, form G(ξ, t) of the Green’s function G(x, t). To bring G β ˆ we shall attempt to use a traditional real analytic method: estimate Dξ (ξ α G(ξ, t)) since β α ˆ β α x Dx G = Dξ (ξ G(ξ, t)) (see Lemma 3.7). Unfortunately, this method does not work for all of the terms in the Green’s function. In Sect. 4, we identify the term that canˆ D (ξ, t) and prove that inside the cone not be estimated by the real analytic method G √ ˆ D behaves like t −3/2 B3/2 (|x|, t), while outside the cone, it behaves |x| ≤ ct + t, G like t −3/2 (1 + t)−1/2 (BN (|x|, t) + BN (|x| − ct, t)). The main difficulty in estimating GD lies in the presence of the double Riesz transform ξj ξk /|ξ |2 . We cannot adopt the
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Table 1. Functions of Technical Lemmas from Sect. 3 Terms
Section
Lemmas from Sect. 3
GD
Sect. 4
3.12, 3.13
Wave Types |α|+3 √ 2 B 3 (|x|, t)χ {|x|≤ct+ t} 2 (|x|−ct)2 |x|2 |α|+3 1 t − 2 (1 + t)− 2 (e− 5t + e− 5t
t−
χ1 (D)GR Sect. 5.1 GR
χ2 (D)GR Sect. 5.2 χ3 (D)GR Sect. 5.3
3.3, 3.4, 3.5, 3.7, 3.8, 3.14
t t
− |α|+3 2 − |α|+3 2
)
BN (|x|, t) 1
(1 + t)− 2 BN (|x| − ct, t)
3.7
|α|+3 t − 2 (1
3.7
t−
|α|+3 2 (1
1
+ t)− 2 BN (|x| − ct, t) 1
+ t)− 2 BN (|x| − ct, t), Fα
traditional strategy because the derivatives of ξj ξk /|ξ |2 behave badly at the origin. We ˆ D (ξ, t). resolve this difficulty by explicitly computing the inverse Fourier transform of G ˆR = G ˆ −G ˆ D into low and high In Sect. 5, we further decompose the remainder G frequency waves. For this, we define the smooth cut-off functions χ1 , χ2 and χ3 such that |χj | ≤ 1 and 1, |ξ | ≤ ι 1, |ξ | > R + 1 χ1 (ξ ) = , χ3 (ξ ) = , χ2 (ξ ) = 1 − χ1 (ξ ) − χ3 (ξ ). 0, |ξ | > 2ι 0, |ξ | < R To mimic the notation for the aforedefined pseudo-differential operator χR (D), we ˆ R (ξ, t). The low frequency let χj (D)GR denote the inverse Fourier transform of χj (ξ )G wave, i.e., χ1 (D)GR (x, t), will be estimated in Sect. 5.1. The middle and high frequency waves will be estimated in Sects. 5.2 and 5.3, respectively. To avoid disrupting the flow of the paper, we put together all the technical lemmas in Sect. 3. The reader should treat this section as a reference section and the functions of these lemmas are summarized in Table 1. 2. Preliminary Computations The Fourier transform of the linearized system (1.2) is 0 −iξ T 0 ρˆ ˆ and M = 0 −|ξ |2 I − ηξ ξ T −2iξ/3 . zˆ t = M zˆ , where zˆ = m ˆ −5iξ T /3 −κ|ξ |2 κ|ξ |2
(2.1)
By using elementary row and column operations, we can compute the eigenvalues of M: −|ξ |2 (with multiplicity 2), λ, λ+ and λ− , and their corresponding eigenvectors: 0 g− 0 g g+ ξ3 0 iξ1 iξ1 iξ1 0 , ξ3 , iξ2 , iξ2 , iξ2 , −ξ −ξ iξ iξ iξ 1 2 3 3 3 0 0 γ γ+ γ− where g=
|ξ |2 |ξ |2 5λ/3 + κ|ξ |2 |ξ |2 5λ± /3 + κ|ξ |2 |ξ |2 , g± = , γ± = ,γ = · · . 2 λ λ± λ + κ|ξ | λ λ± + κ|ξ |2 λ±
(2.2)
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D.L. Li
Here λ and λ± are the roots of the equation λ3 + (κ + ν)|ξ |2 λ2 + (νκ|ξ |4 +
10 2 2 |ξ | )λ + κ|ξ |4 = 0, 9 3
(2.3)
where ν = + η. While explicit formulas for λ and λ± exist, they are too complicated to be of any practical use. We can, however, obtain the estimates of these roots by using the Implicit Function Theorem. Theorem 2.1. For sufficiently small |ξ |, λ is real and λ± are complex conjugates. Furthermore, ∞
3 aj |ξ |2j ; a. λ = − κ|ξ |2 + 5
b. λ± = −
j =2
1 1 κ + ν |ξ |2 + 5 2
√ 10/9 and a¯ j ’s are real.
∞
a¯ 2j |ξ |2j ± i c|ξ | +
j =2
∞
a¯ 2j −1 |ξ |2j −1 , where c =
j =2
Proof. We shall only prove (b) because the proof for (a) is analogous. Let r = |ξ | and consider the equation r 3 · (λ¯ 3 + (κ + ν)r λ¯ 2 + (νκr 2 + 10/9)λ¯ + 2κr/3) = 0.
(2.4)
¯ = λ¯ 3 + (κ + ν)r λ¯ 2 + (νκr 2 + 10/9)λ¯ + Define F : C × C →√ C as follows: F (r, λ) √ 2κr/3. Note that F (0, i 10/9) = 0 and Fλ¯ (0, i 10/9) = 0. The Implicit Function ¯ Theorem√ implies that for sufficiently small r, there exists an analytic function λ(r) with ¯ ¯ ¯ λ(0) = i 10/9 and F (r, λ(r)) = 0. Therefore λ(r) must be a root of (2.4). Write ¯ λ(r) = a˜ 1 + a˜ 2 r + a˜ 3 r 2 + · · · , where a˜ j +1 =
√ λ¯ j (0) ¯ = i 10/9, i.e., . In particular, a˜ 1 = λ(0) j! ¯ λ(r) = i 10/9 + a˜ 2 r + a˜ 3 r 2 + · · · .
¯ Then (2.4) becomes Now let λ+ (r) = r λ(r). λ+ (r)3 + (κ + ν)r 2 λ+ (r)2 + (νκr 4 +
10 2 2 r )λ+ (r) + κr 4 = 0. 9 3
Thus λ+ (r) is a root of (2.3) and for sufficiently small |ξ |, λ+ = r λ¯ = i 10/9|ξ | + a˜ 2 |ξ |2 + a˜ 3 |ξ |3 + · · · . Similarly we can prove that λ− = r λ¯ = −i 10/9|ξ | + a˜ 2 |ξ |2 + a˜ 3 |ξ |3 + · · · . Now it remains to show that the real part of λ± contains only even powers of |ξ | and the imaginary part of λ± only odd powers of |ξ |. Since (2.3) is an equation of order
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
585
3 with real coefficients, we know that λ± must be complex conjugates. Furthermore, λ + λ+ + λ− = −(κ + ν)|ξ |2 . So
∞ 1 1 1 1 2 2 aj |ξ |2j . κ + ν |ξ | − Re λ+ = Re λ− = (−(κ + ν)|ξ | − λ) = − 2 5 2 2 j =2
To show that the imaginary part of λ± contains only odd powers of |ξ |, note that λλ+ λ− = −2κ|ξ |4 /3 ⇒ λ+ λ− = −2κ|ξ |4 /3λ. On the other hand, since λ+ and λ− are conjugates, λ+ λ− = (Re λ+ )2 +(Im λ+ )2 . Hence (Im λ+ )2 = −2κ|ξ |4 /3λ−(Re λ+ )2 . Note that the right-hand side contains only even powers of |ξ |. We already know that the lowest term of Im λ+ is c|ξ |. Now assume that Im λ+ contains even powers of |ξ | and the lowest such term with non-zero coefficient is a˜ 2j |ξ |2j . Then (Im λ+ )2 contains odd powers of |ξ | and the lowest such term is 2ca˜ 2j |ξ |2j +1 , which is non-zero by our assumption. However, (Im λ+ )2 contains only even powers of |ξ | as argued above, which is a contradiction. Thus, the imaginary part of λ± contains only odd powers of |ξ | and this completes the proof. Theorem 2.2. For 0 < ι ≤ |ξ | ≤ R, there exists b > 0 such that Re(λ, λ± ) ≤ −b. Proof. Let f (x) = x 3 + (κ + ν)|ξ |2 x 2 + (νκ|ξ |4 +
Fix 0 < b1 < min 1,
10 2 2 |ξ | )x + κ|ξ |4 . 9 3
2κι4 /3 . We shall first show that f (x) > 0 for νκR 4 + 10R 2 /9 + 1 any x ≥ −b1 . Observe that f (x) ≥ −b13 − (νκ|ξ |4 + 10|ξ |2 /9)b1 + 2κ|ξ |4 /3 ≥ −(νκ|ξ |4 + 10|ξ |2 /9 + 1)b1 + 2κ|ξ |4 /3 2κι4 /3 > −(νκR 4 + 10R 2 /9 + 1) · + 2κ|ξ |4 /3 νκR 4 + 10R 2 /9 + 1 > 0. Therefore all real roots of f (x) = 0 are less than −b1 . If λ and λ± are all real, then the theorem is proved. Otherwise, there must be a real root λ and two complex conjugate roots λ± . By the above argument, we already know λ ≤ −b1 . Write λ± = Reλ+ ±iImλ+ , we just need to show Reλ+ is also bounded away from zero. Note that −(κ + ν)|ξ |2 = λ + λ+ + λ− = λ + 2Reλ+ . Therefore 2Reλ+ = −(κ + ν)|ξ |2 − λ. Rewrite f (x) as follows, f (x) = f (−(κ + ν)|ξ |2 ) + f (−(κ + ν)|ξ |2 )(x + (κ + ν)|ξ |2 ) f (−(κ + ν)|ξ |2 ) + (x + (κ + ν)|ξ |2 )2 + (x + (κ + ν)|ξ |2 )3 . 2 Note that f (−(κ + ν)|ξ |2 ) = −(κ + ν)νκ|ξ |6 − 10(κ + ν)|ξ |4 /9 + 2κ|ξ |4 /3 ≤ −(κ + ν)νκι6 , f (−(κ + ν)|ξ |2 ) = ((κ + ν)2 + νκ)|ξ |4 +10|ξ |2 /9 ≤ ((κ + ν)2 + νκ)R 4 + 10R 2 /9, f (−(κ + ν)|ξ |2 ) = −4(κ + ν)|ξ |2 < 0.
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D.L. Li
(κ + ν)νκι6 ), then for any −(κ + ((κ + ν)2 + νκ)R 4 + 10R 2 /9 + 1 2 2 ν)|ξ | ≤ x ≤ −(κ + ν)|ξ | + b2 , we have Now fix 0 < b2 < min(1,
f (x) ≤ −(κ + ν)νκι6 + [((κ + ν)2 + νκ)R 4 + 10R 2 /9]b2 + b23 ≤ −(κ + ν)νκι6 + [((κ + ν)2 + νκ)R 4 + 10R 2 /9 + 1]b2 < 0. Thus, the real root λ must satisfy either λ > −(κ + ν)|ξ |2 + b2 or λ < −(κ + ν)|ξ |2 . But as previously indicated, f (−(κ + ν)|ξ |2 ) < 0, f (b1 ) > 0 and there is only one real root. Therefore, we must have λ > −(κ + ν)|ξ |2 + b2 . This implies that Reλ+ ≤ −b2 /2, which completes the proof. Theorem 2.3. For sufficiently large |ξ |, λ and λ± are all real. Furthermore, ∞
a. λ = −
2 aj |ξ |−2j ; + 3ν j =1
b. λ+ = −ν|ξ |2 + c. λ− = −κ|ξ |2 +
∞ j =0 ∞
aj+ |ξ |−2j ; aj− |ξ |−2j .
j =0
Proof. We omit the proof because it is similar to that of Theorem 2.1.
To complete this section, we shall now compute the Green’s function for the lineˆ 0, ˆ 0 )T . The solution to (2.1) can be arized system (1.2) with initial data zˆ 0 = (ρˆ0 , m written as follows: ρˆ 0 0 ˆ ξ m 0 2 1 3 ˆ 2 = A 0 + B ξ3 e−|ξ | t m m −ξ1 ˆ3 −ξ2 ˆ 0 0 g g+ g− iξ1 iξ1 iξ1 + C iξ2 eλt + D iξ2 eλ+ t + E iξ2 eλ− t . (2.5) iξ iξ iξ 3 3 3 γ γ+ γ− Using the initial values, we have
0 0 0 ξ3 0
ξξT A 0 + B ξ3 = m ˆ0 I− , 2 −ξ −ξ |ξ | 1 2 0 0 0
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
587
ξT m ˆ0 + C3 ˆ 0, |ξ |2 ξT m ˆ0 + D3 ˆ 0, D = D1 ρˆ0 + iD2 |ξ |2 ξT m ˆ0 + E3 ˆ 0. E = E1 ρˆ0 + iE2 |ξ |2
C = C1 ρˆ0 + iC2
Here γ− − γ+ C1 = γ − γ− D1 = γ −γ E1 = +
γ− g+ − γ+ g− C2 = γ g− − γ − g D2 = γ g − γ g+ E2 = +
g+ − g− C3 = g− − g D3 = g − g+ E3 =
, (2.6)
where = −γ− g+ +γ− g+γ g+ +γ+ g− −γ+ g−γ g− . In this notation, the Fourier transˆ + eλ+ t + G ˆ − eλ− t , ˆ =G ˆ∗ +G form of the Green’s function for (1.2) can be written as G where T λt λt iξ λt e gC e · gC e gC 1 2 3 2
|ξ | T T ξ ξ λt λt , ˆ ∗ = iξ C1 eλt e−|ξ |2 t I − ξ ξ G − C e iξ C e 2 3 2 2 |ξ | |ξ | T λt λt iξ λt γ C1 e γ C2 e · 2 γ C3 e |ξ |
iξ T D g D · g D g + 1 + 2 |ξ |2 + 3 T ˆ + = iξ D1 −D2 ξ ξ G iξ D 3 , 2 |ξ | iξ T γ+ D 1 γ+ D 2 · 2 γ+ D 3 |ξ |
(2.7)
iξ T E g E · g E g − 1 − 2 |ξ |2 − 3 T ˆ − = iξ E1 −E2 ξ ξ G iξ E 3 . (2.8) 2 |ξ | iξ T γ − E 1 γ − E 2 · 2 γ − E3 |ξ |
3. Useful Lemmas In this section we establish some lemmas that we shall use in Sects. 4 and 5. Lemma 3.1 is basically Kirchhoff’s formula and its proof can be found in [5]. We omit the proofs for Lemmas 3.2 and 3.3 since they are standard results in calculus. Lemmas 3.7 and 3.8 establish that BN behaves much like the heat kernel and is of crucial importance in estimating GR (x, t). Notation. Let α = (α 1 , . . . , α n ) and β = (β 1 , . . . , β n ) be multi-indices. We say α ≤ β if and only if α j ≤ β j for all 1 ≤ j ≤ n, and α < β if and only if α ≤ β and α j < β j for some j . When decomposing a multi-index, we write β = βj , where each βj is a multi-index. Subscripts are used here to avoid confusion.
588
D.L. Li
Lemma 3.1. Let ξ ∈ Rn where n is odd and wˆ = (2π )−n/2 (sin c|ξ |t)/(c|ξ |). Then for any smooth function f there are constants aη , bη such that aη t |η|+1 D η f (x + cty)y η dSy , (3.1) w∗f = 0≤|η|≤(n−3)/2
wt ∗ f =
bη t |η|
|y|=1
0≤|η|≤(n−1)/2
|y|=1
D η f (x + cty)y η dSy .
(3.2)
Lemma 3.2 (Chain Rule). Let g(ξ ) = F (h(ξ )). Then β Dξ g(ξ )
=
|β|
j
F (h(ξ ))
j =1
|η k |≥1 j
ηk =β
j
η
Dξ k h(ξ ).
k=1
k=1
Lemma 3.3. (2k)
a. Dξj |ξ |l =
k
Hi ξj2i |ξ |l−2k−2i , where Hi = Hi (k, l).
i=0 k |ξ |l = Hi ξj2i+1 |ξ |l−2k−2i−2 , where Hi = Hi (k, l). i=0 β β c. If m is odd, Dξ |ξ |m ≤ C|ξ |m−|β| , where C = C(β, m). If m is even, Dξ |ξ |m ≤ (2k+1)
b. Dξj
β
C|ξ |m−|β| for |β| ≤ m and when |β| > m, Dξ |ξ |m = 0. |βj |≥1 βj β f (ξ )t (D f (ξ ))t ef (ξ )t . d. Dξ e = ξ
βj =β
βj
Lemma 3.4. Suppose that for sufficiently small |ξ |, f (ξ ) = Then
k β f (ξ )t t + ≤ Cβ Dξ e 1≤k< |β| 2
Proof. By Lemma 3.2, we have
∞ j =2
fj |ξ |j , where fj ∈ C.
|ξ |2k−|β| t k ef (ξ )t .
|β| 2 ≤k≤|β|
|≥1 k |β| |β βj j β f (ξ )t k f (ξ )t f (ξ ) t e ≤ D . Dξ e ξ k=1 k j =1 j =1
βj =β
∞ β Since for sufficiently small |ξ |, f (ξ ) = fj |ξ |j , we have Dξ j f (ξ ) ≤ Cβj for j =2 βj |βj | ≥ 2, while for |βj | = 1, we have Dξ f (ξ ) ≤ Cβj |ξ |.
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
Claim. If
k j =1
589
βj = β, k ≥ |β|/2, then there are at least (2k − |β|) βj ’s with |βj | = 1.
Proof. Assume that there are k βj ’s with |βj | = 1. Then there are (k − k ) βj ’s with |βj | ≥ 2. Note that |β| =
k
|βj | =
|βj | +
|βj |=1
j =1
|βj | ≥ k + 2(k − k ) = 2k − k ⇒ k ≥ 2k − |β|.
|βj |≥2
By the claim, therefore, we have
k β j Dξ f (ξ ) ≤ Cβ |ξ |2k−|β| for k ≥ |β|/2. Thus
j =1
k β f (ξ )t t + ≤ Cβ Dξ e 1≤k< |β| 2
|ξ |2k−|β| t k ef (ξ )t .
|β| 2 ≤k≤|β|
Lemma 3.5. Suppose that for sufficiently small |ξ |, f (ξ ) = a2 < 0 and l > −n. Then |ξ | small
|ξ |l ef (ξ )t dξ ≤ Cl,n t −
l+n 2
∞ j =2
aj |ξ |j , where ξ ∈ Rn ,
.
Proof. Note that ef (ξ )t = e Therefore,
|ξ | small
∞
j j =2 aj |ξ | t
|ξ |l eλt dξ ≤ Ct −
≤ e−δ|ξ | t for some δ > 0.
l+n 2
2
R
The last integral converges because l > −n.
s
l+n−2 2
e−δs ds ≤ Cl,n t −
l+n 2
.
Lemma 3.6 [5]. For 2N ≥ n, we have n−1 I= B2N (x + cty, t)dSy ≤ C(1 + t)− 2 BN (|x| − ct, t). |y|=1
Proof. Notice that I is rotation invariant, so without loss of generality, we can assume that x = (|x|, 0, . . . , 0). Then |x + cty|2 = |x|2 + 2ct|x|y1 + c2 t 2 |y|2 . Write B2N (x + cty, t)dSy = I1 + I2 , |y|=1
where I1 =
|y|=1,y1 <0
B2N (x + cty, t)dSy ,
I2 =
|y|=1,y1 ≥0
B2N (x + cty, t)dSy .
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D.L. Li
Clearly, I2 ≥ I1 , so we only need to estimate I1 . Note that |x + cty|2 = |x|2 + 2ct|x|y1 + c2 t 2 |y|2 = (|x| − ct)2 + 2c2 t 2 (1 + y1 ) + 2ct (|x| − ct)(1 + y1 ). Since 2ct (|x| − ct)(1 + y1 ) ≥ − 23 (|x| − ct)2 (1 + y1 ) − 23 c2 t 2 (1 + y1 ) and y1 < 0, we have 1 1 |x + cty|2 ≥ (|x| − ct)2 + c2 t 2 (1 + y1 ). 3 2 Since −1 ≤ y1 < 0, we know that c2 t 2 (1 + y1 ) ≥ 0. Let P 2 = (x + cty)2 /t, Q2 = (|x|−ct)2 /(3t) and R 2 = c2 t (1+y1 )/2. From the above, we have Q2 +R 2 ≤ P 2 . This also implies that Q2 ≤ P 2 and R 2 ≤ P 2 , thus we have Q2 R 2 ≤ P 4 . Therefore, (1 + Q2 )(1 + R 2 ) ≤ 1 + P 2 + P 4 ≤ (1 + P 2 )2 . Thus
−2N
−N
−N |x + cty|2 c2 t (1 + y1 ) (|x| − ct)2 1+ 1+ ≤ 1+ . t 3t 2
Hence,
I1 ≤ 3N BN (|x| − ct, t)
|y|=1,y1 <0
1+
c2 t (1 + y1 ) 2
−N dSy .
To complete the proof, we just need to show that the above integral is bounded by (1 + t)−(n−1)/2 . For t ≤ 1, we know that
−N n−1 c2 t (1 + y1 ) 1+ dSy ≤ Cn ≤ Cn (1 + t)− 2 . (3.3) 2 |y|=1,y1 <0 For t > 1, we decompose the integral into two parts. Consider y1 ≥ −1/2. Since 2N ≥ n − 1, we have
−N n−1 c2 t (1 + y1 ) 1+ dSy ≤ Cn (1 + t)−N ≤ Cn (1 + t)− 2 . 1 2 |y|=1,− 2 ≤y1 <0 n−1 Now if y1 < −1/2, we let w = (y2 , . . . , yn ) ∈ R . Since |y| = 1, we have y1 = 2 − 1 − |w| and
|w|2 |w|2 ≥ . 2 1 + 1 − |w|2 √ Also, y1 < −1/2 and |y| = 1 implies that |w| ≤ 3/2. Therefore,
−N −N c2 t (1 + y1 ) c2 t|w|2 1+ 1+ dSy ≤ dw √ 2 4 |y|=1,y1 <− 21 |w|≤ 23 −N ∞
c2 s 2 − n−1 1+ s n−2 ds ≤ CN t 2 4 0 1 + y1 = 1 −
1 − |w|2 =
≤ CN (1 + t)−
n−1 2
.
We have used the fact t > 1 and the last integral converges because 2N ≥ n.
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
591
Lemma 3.7. Let N be an integer and suppose that for any multi-index |β| ≤ N and any x ∈ Rn , fˆ(ξ, t) satisfies |α|+n+k ix·ξ 2β α ˆ e D ξ f (ξ, t) dξ ≤ Cα,β t − 2 t |β| . ξ Rn
|α|+n+k Then Dxα f (x, t) ≤ Cα,N t − 2 BN (|x|, t). 2β Proof. Observe that if we let g(x, t) = x 2β Dxα f (x, t), then g(ξ, ˆ t) = Cn Dξ (ξ α fˆ(ξ, t)). Therefore, for any multi-index β, |α|+n+k 2β α ix·ξ 2β α ˆ ≤ Cα,β t − 2 t |β| . (3.4) D f (x, t) = C e D (ξ f (ξ, t))dξ x n x ξ
Rn
Now fix x and without loss of generality, we assume that x1 = |x|∞ , where | · |∞ denotes the sup norm on Rn . Since (3.4) holds for any |β| ≤ N , we choose β = (N, 0, . . . , 0) so that |x 2β | = |x|2N ∞ . Hence, α |α|+n+k |α|+n+k D f (x, t) ≤ Cα,N t − 2 t N |x|−2N = Cα,N t − 2 t N |x|−2N . x ∞ The last inequality follows from the equivalency of the sup norm and the Euclidean |α|+n+k norm on Rn . Also, if we let β = 0, then Dxα f (x, t) ≤ Ct − 2 . Therefore,
α |α|+n+k t N D f (x, t) ≤ Cα,N t − 2 min 1, . x |x|2 Since 1+
|x|2 t
1, |x|2 ≤ t ≤ 2 |x|2 , , |x|2 > t t
we have
|x|2 1+ t
−N ≥2
−N
1,
t |x|2
N
|x|2 ≤ t , |x|2 > t
.
−N
α |α|+n+k |x|2 t N − 2 − |α|+n+k 2 Thus Dx f (x, t) ≤ Cα,N t ≤ Cα,N t 1+ min 1, . |x|2 t |α|+n+k Lemma 3.8. Suppose that Dxα f (x, t) ≤ Cα,N t 2 BN (|x|, t) for any N . Then α |α|+n+k D wt ∗ f (x, t) ≤ Cα,N t − 2 (1 + t)− n−1 4 BN (|x| − ct, t), x α |α|+n+k−1 n−1 D w ∗ f (x, t) ≤ Cα,N t − 2 (1 + t)− 4 BN (|x| − ct, t). x
(3.5) (3.6)
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D.L. Li
Proof. By Lemmas 3.1 and 3.6, we have
α D (wt ∗ f (x, t)) ≤ Cα,N x
bη t |η| t −
|α|+|η|+n+k 2
(1 + t)−
n−1 2
BN (|x| − ct, t)
0≤|η|≤(n−1)/2
≤ Cα,N t −
|α|+n+k 2
(1 + t)−
n−1 4
BN (|x| − ct, t).
The second inequality (3.6) can be proved in a similar way. α − |x|2 |α| |x|2 ≤ Cα,a s − 2 e− as for any a > 4. 4s Lemma 3.9. Dx e Proof. By Lemma 3.2, we have
α − |x|2 D e 4s ≤ C x |α| ≤k≤|α| 2
|αj |≥1
k j =1
≤ Cα |α|
j =1
k− |x|2
|α| 2
s
|x|2 as
|αj |
s
e
2
−k − |x| 4s
αj =α
|α| 2 ≤k≤|α|
≤ Cα,a s − 2 e−
|x|
2k−
k
2 − |α| − |x| s 2 e 4s
. α
We have used the fact that |αj | = 1 or 2 since Dx j (|x|2 /4s) = 0 for |αj | > 2.
√ 3 Lemma 3.10. If |x| ≥ δ t for some fixed δ ≥ 0, then |x|−3 ≤ C(δ)t − 2 B 3 (|x|, t). 2
3
3
Proof. |x|−3 ≤ (1 + δ 2 ) 2 (δ 2 t + δ 2 |x|2 )− 2 ≤ Lemma 3.11. Let f (r) =
ct+r
1 + δ2 δ2
23
3
t− 2
1+
|x|2 t
− 23 .
s2
e− 4t ds, then for any k ≥ 0,
ct−r
a.
f 2k (0)
= 0;
b. f 2k+1 (0) =
k
c2 t
m,k t −m e− 4 .
m=0 s2
s2
s2
e− 4t ds + 0 e− 4t ds. Let h(r) = 0 e− 4t ds, Proof. Note that f (r) = 0 then f (r) = h(r) − h(−r), which implies f 2k (0) = 0. As for f 2k+1 (0), we first notice that r+ct
h2k+1 (r) =
r−ct
d 2k d 2k − (r+ct)2 h (r) = e 4t . dr 2k dr 2k
r+ct
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
593
By Lemma 3.2, 2k (r+ct)2 d 2k − (r+ct)2 4t e = e− 4t 2k dr m=1
|η l |≥1 m
ηl =2k
m (r + ct)2 d ηl − . dr ηl 4t l=1
l=1
(r + ct)2 d ηl Note that in the above summation, ηl ≤ 2 because otherwise η − dr l 4t = 0. Therefore, when r = 0, the power of t goes from 0 (when ηl = 1 for all l) to −k (when ηl = 2 for all l). Hence, k c2 t h2k+1 (0) = m,k t −m e− 4 . m=0
This completes the proof since f 2k+1 (0) = 2h2k+1 (0).
Lemma 3.12. For any multi-index α,
|x| α ∂2 |α|+2 s2 1 − 4t ≤ Cα t − 2 B 3 (|x|, t). D e ds x ∂x ∂x 2 |x| 0 j k Proof. We divide the proof into two cases. s2
Case 1: |x|2 < δt. Since |x|2 /t is sufficiently small in this case, we can expand e− 4t in Taylor series:
|x| α ∂2 s2 1 − 4t D e ds x ∂x ∂x |x| j k 0 ∞ 2 2m+1 1 1 |x| α ∂ = Dx (−1)m ∂xj ∂xk |x| (2m + 1) · m! (4t)m m=0
2 m− |α|+2 ∞ 2 1 |x| − |α|+2 ≤ Cα t 2 t ! " (2m + 1) · m! m=
≤ Cα t −
|α|+2 2
|α|+2 2
.
Notice that B 3 (|x|, t) ≥ 1/(1 + δ)3/2 when |x|2 < δt, so the lemma is proved in this 2 case. |x| s2 ∂ xk − |x|2 Case 2: |x|2 ≥ δt. First we have e 4t . Therefore, for any e− 4t ds = ∂xk 0 |x| |α| ≥ 1,
α D x
|x|
2
e 0
s − 4t
α−e xk − |x|2 k e 4t ≤ ds = Dx |x|
α1 +α2 =α−ek
α xk α − |x|2 D 1 D 2 e 4t . x |x| x
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D.L. Li
|α1 | xk By Lemma 3.3, Dxα1 ≤ C|x|−|α1 | ≤ Ct − 2 . Also, by Lemma 3.9, |x| α − |x|2 |α2 | |x|2 − − D 2 e 4t ≤ Cα t 2 e 5t . Thus, x
α D x
|x| 0
s2 e− 4t ds ≤ Cα
t−
|α1 | 2
· t−
|α2 | 2
|x|2
e− 5t ≤ Cα t −
|α|−1 2
|x|2
e− 5t .
α1 +α2 =α−ek
Hence,
|x| α ∂2 s2 1 − 4t D e ds x ∂x ∂x |x| j k 0
|x| α ∂2 s2 1 e− 4t ds ≤ Dx ∂xj ∂xk |x| 0
|x| |α 2 |≥1 α s2 − 4t D 1 1 D α2 + e ds x |x| x 0 α +α =α+e +e 1
2
j
k
1
≤ Cα |x|−|α|−3 t 2
|x| √ t
|α 2 |≥1
u2
e− 4 du + Cα
0
|x|−|α1 |−1 t −
|α2 |−1 2
|x|2
e− 5t
α1 +α2 =α+ej +ek
1
≤ Cα |x|−|α|−3 t 2 + Cα t
− |α|+2 2
e
2 − |x| 5t
≤ Cα t −
|α|+2 2
B 3 (|x|, t). 2
We have used Lemma 3.10 in the last step.
Lemma 3.13. For any multi-index α,
ct+|x| 2 α ∂2 s2 1 − 4t − |α|+2 − 21 − (|x|−ct) D 2 5t B 3 (|x|, t) + (1 + t) e . e ds ≤ Cα t x ∂x ∂x 2 |x| ct−|x| j k Proof. We divide the proof into six cases. Case 1: |x| ≤ δ, t ≤ δ, |x|2 ≤ δt. For sufficiently small |x|, we can use Lemma 3.11 to ct+|x| s2 e− 4t ds in Taylor series, expand ct−|x|
ct+|x|
2
e
s − 4t
ct−|x|
Therefore,
α D x
∞ 1 ds = k! k=0
∂2 ∂xj ∂xk
1 |x|
! k=
ct+|x|
ct−|x|
∞
≤ Cα
≤ Cα t
|α|+2 2
! k=
m,k t
−m
c2 t
e− 4 |x|2k+1 .
m=0
s2 e− 4t ds
2 k |x|2k−(|α|+2) e− c4t Since t ≤ δ tk " k!
∞
− |α|+2 2
k
|α|+2 2
k " k!
|x|2 t
k− |α|+2 2 .
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The infinite sum in the last step converges because |x|2 /t is sufficiently small. We also have that (1 + δ)−3/2 ≤ (1 + |x|2 /t)−3/2 . Thus α D x
∂2 ∂xj ∂xk
1 |x|
ct+|x|
ct−|x|
|α|+2 s2 e− 4t ds ≤ t − 2 B 3 (|x|, t). 2
Case 2: |x| ≤ δ, t ≤ δ, |x|2 > δt. First we have
∂ ∂xk
ct+|x|
2
e
s − 4t
ct−|x|
2 (|x|−ct)2 xk − (|x|+ct) − 4t 4t . e ds = +e |x|
Therefore, for any |α| ≥ 1,
α D x
ct+|x| ct−|x|
(|x|+ct)2 (|x|−ct)2 s2 xk e− 4t ds = Dxα−ek e− 4t + e− 4t |x|
≤ Cα
|x|−|α1 | t −
|α2 | 2
e−
(|x|−ct)2 5t
α1 +α2 =α−ek
≤ Cα t −
|α|−1 2
e−
(|x|−ct)2 5t
.
Next, for any multi-index α, we have α D x
ct+|x| s2 ∂2 1 − 4t e ds ∂xj ∂xk |x| ct−|x|
ct+|x| s2 ∂2 1 e− 4t ds ≤ Dxα ∂x ∂x |x| j
k
ct−|x|
|α 2 |≥1
+
α1 +α2 =α+ej +ek
≤ Cα t −
|α|+2 2
α D 1 1 D α2 x |x| x
B 3 (|x|, t) + Cα t −
|α|+2 2
2
e−
ct+|x| ct−|x|
(|x|−ct)2 5t
s2 e− 4t ds
.
We have used Lemma 3.10 in the last step. Finally, (1 + δ)−1/2 ≤ (1 + t)−1/2 since t ≤ δ. Therefore, α D x
∂2 ∂xj ∂xk
≤ Cα t −
|α|+2 2
1 |x|
ct+|x|
2
e
s − 4t
ct−|x|
ds
B 3 (|x|, t) + Cα t − 2
|α|+2 2
1
(1 + t)− 2 e−
(|x|−ct)2 5t
.
Notice that in this case we only needed the hypotheses |x|2 ≥ δt and t ≤ δ. Case 3: |x| ≤ δ, t > δ. As in Case 1, we have
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D.L. Li
ct+|x| α ∂2 s2 1 − 4t D e ds x ∂x ∂x |x| ct−|x| j k ∞ k 2 1 2 α ∂ −m − c4t 2k = Dx e m,k t |x| ∂xj ∂xk k! m=0
k=0
≤ Cα e
2 − c4t
≤ Cα t
− |α|+2 2
B 3 (|x|, t). 2
The last step follows from 2
e
− c4t
≤e
−bt
(t +
√
δ)
− 23
≤e
−bt − 23
t
|x|2 1+ t
− 23 .
Case 4: |x| > δ, |x| ≤ ct/2. As in Case 2, we have for any |α| ≥ 1,
ct+|x| 2 2 α |α | s2 − 4t −|α1 | − 22 − 6(|x|−ct) − 6(|x|−ct) D ≤ Cα 25t 25t e ds |x| t e ≤ C e . α x ct−|x|
α1 +α2 =α−ek
The last inequality follows from the fact that both |x| and t are strictly bounded away from zero in this case. Also, |x| ≤ ct/2, thus we have e−
6(|x|−ct)2 25t
and
= e−
ct+|x|
(|x|−ct)2 25t
e−
(|x|−ct)2 5t
c2 t
≤ e− 100 e−
s2
e− 4t ds ≤ 2|x|e−
(|x|−ct)2 4t
(|x|−ct)2 5t
≤ (1 + t)−N e−
≤ C(1 + t)−N e−
(|x|−ct)2 5t
(|x|−ct)2 5t
ct−|x|
for any N . Therefore, for any multi-index α,
ct+|x| α ∂2 s2 1 − 4t D e ds x ∂x ∂x |x| ct−|x| j k
ct+|x| α ∂2 s2 1 e− 4t ds ≤ Dx ∂x ∂x |x| j
k
ct−|x|
|α 2 |≥1
+
α1 +α2 =α+ej +ek
≤ (1 + t)−N e−
α D 1 1 D α2 x |x| x
(|x|−ct)2 5t
ct+|x|
2
e ct−|x|
s − 4t
ds
.
Case 5: |x| > δ, |x| > ct/2, t ≥ δ. As in Case 2, for any |α| ≥ 1,
ct+|x| 2 α |α | s2 − 4t −|α1 | − 22 − (|x|−ct) D ≤ Cα 5t e ds |x| t e . x ct−|x|
α1 +α2 =α−ek
Since |x| ≥ ct/2 and t ≥ δ, we have
ct+|x| α s2 − 4t D e ds ≤ Cα x ct−|x|
α1 +α2 =α−ek
t −|α1 | t −
|α2 | 2
e−
(|x|−ct)2 5t
≤ Cα t −
|α|−1 2
e−
(|x|−ct)2 5t
.
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Notice that t ≥ δ implies that t −1 ≤ Ct −1/2 and t −1 ≤ C(1 + t)−1 . Therefore,
ct+|x| α ∂2 s2 1 − 4t D e ds x ∂x ∂x |x| ct−|x| j k
∂2 1 ct+|x| − s 2 e 4t ds ≤ Dxα ∂x ∂x |x| j
k
ct−|x|
|α 2 |≥1
+
α1 +α2 =α+ej +ek |α|+2 2
= Cα t −
α D 1 1 D α2 x |x| x
B 3 (|x|, t) + Cα t −
|α|+2 2
2
ct+|x| ct−|x| 1
(1 + t)− 2 e−
s2 e− 4t ds
(|x|−ct)2 5t
.
Case 6: |x| > δ, |x| > ct/2, t < δ. We have |x|2 > δ 2 > δt and t < δ. Therefore, the proof is reduced to that of Case 2. Lemma 3.14. Suppose that F1 (x) = 0 sin(|β|) (s)(x−s)|β|−1 ds and F2 (x) = 0 cos(|β|) (s) (x − s)|β|−1 ds, where |β| ≥ 1, then for sufficently small |ξ | and for all |β2 | ≤ |β|, we have x
x
|Dξ 2 Fi (Imτ t)| ≤ Cβ |ξ |3|β|−|β2 | t |β| , i = 1, 2. β
Proof. We shall only prove the lemma for F1 since the proof for F2 is analogous. Recall x x ∂ that if F (x) = 0 H (x, s)ds, then F (x) = H (x, x) + 0 ∂x H (x, s)ds. Therefore, for j < |β|, x (j ) F1 (x) = Cj sin(|β|) (s)(x − s)|β|−1−j ds. 0
(|β|)
When j = |β|, we have F1
(j ) (x) = Cβ sin(|β|) (x). Therefore, for all j ≤ |β|, F1 (x) ≤
Cβ x |β|−j . Now, for any β2 ≤ β, by Lemma 3.2 we have |β2 | 3 |β|−j β2 |ξ | t Dξ F1 (Imτ t) ≤ Cβ j =1
|η k |≥1 j
|ξ |
3j −
j
k=1
|ηk |
t j ≤ Cβ |ξ |3|β|−|β2 | t |β| .
ηk =β2
k=1
Lemma 3.15. Suppose that F (x) = 0 es (x − s)|β|−1 ds, where |β| ≥ 1, then for sufficently small |ξ | and for all |β2 | ≤ |β|, we have ∞ ∞ 2k β2 2k a¯ 2k |ξ | t ≤ Cβ |ξ |4|β|−|β2 | t |β| e k=2 a¯ 2k |ξ | t . Dξ F x
k=2
Proof. We omit the proof since it is analogous to that of Lemma 3.14.
4. Pointwise Estimates for GD (x, t) In this section, we shall first identify the term that cannot be estimated by the real analytic method mentioned in Sect. 1. Define:
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D.L. Li
0
0
0
T ˆ D = 0 ξ ξ (cos(c|ξ |t) − 1)e−|ξ |2 t ) 0 . G |ξ |2 0 0 0
(4.1)
ˆ The justification of the above definition lies in the structure of G(x, t) and its behavior when |ξ | is small. This will become evident in Sect. 5.1. For now, we shall proceed ˆD to estimate GD . The following lemma computes the inverse Fourier transform of G explicitly.
1
(|x|−ctr)2 (|x|+ctr)2 1 1 ∂2 jk − − 4t 4t e dr for −e Lemma 4.1. GD (x, t) = Ct 2 ∂xj ∂xk |x| 0 2 ≤ j, k ≤ 4. Proof. First we define g(ξ, ˆ t) as follows: g(ξ, ˆ t) =
(cos(c|ξ |t) − 1) −|ξ |2 t e = −c2 |ξ |2
t 0
s
cos(c|ξ |u)e−|ξ | t duds. 2
0
Since g(ξ, ˆ t) is Schwarz, we can use the Inversion Formula, t s 2 ix·ξ e cos(c|ξ |u)e−|ξ | t dudsdξ. g(x, t) = C R3
0
0
−|ξ |2 u
Note that eix·ξ cos(c|ξ |u)e ∈ L1 (R3 × ), where denotes the triangular region in the integral duds. We can apply Fubini, t s 2 eix·ξ cos(c|ξ |u)e−|ξ | t dξ duds. g(x, t) = C 0
R3
0
Now apply Kirchhoff’s formula, g(x, t) = C
t 0
s 0
∂ u ∂u
|z|2 3 − t − 2 e− 4t dSz duds.
|z−x|=cu
Since u
|z|2 3 − t − 2 e− 4t dSz is differentiable with respect to u and is equal to 0 at
|z−x|=cu
u = 0, we have by the Fundamental Theorem of Calculus, t |z|2 3 1 g(x, t) = C s − t − 2 e− 4t dSz ds = t 2 0
|z−x|=cs
|y|≤1
1 − |x+cty|2 e 4t dy. |y|
Therefore, jk
1
GD (x, t) = Ct 2
∂2 ∂xj ∂xk
|y|≤1
1 − |x+cty|2 e 4t dy. |y|
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2
Notice that the integral |y|≤1
1 − |x+cty| 5t |y| e
dy is rotation invariant with respect to x, so
without loss of generality, we can assume x = (|x|, 0, 0), 2π π 1 1 − |x+cty|2 1 − |x|2 +c2 t 2 r 2 +2|x|ctr cos φ 2 4t r sin φdrdφdθ e 4t dy = e |y| 0 0 0 r |y|≤1
1 2cr|x| cos φ φ=π 4 e re · dr =C r|x| 0 φ=0 1
(|x|−ctr)2 (|x|+ctr)2 1 e− 4t − e− 4t dr. =C |x| 0
1
2 +c2 t 2 r 2 4t
− |x|
Note that 1
e−
(|x|−ctr)2 4t
− e−
(|x|+ctr)2 4t
dr =
0
1 ct
|x|
|x|−ct
s2
e− 4t ds −
|x|+ct |x|
s2
e− 4t ds
|x| ct+|x| s2 s2 1 − 4t − 4t 2 e ds − e ds . = ct 0 ct−|x|
Therefore, by Lemmas 3.12, 3.13 and 4.1, we have
2 α |α|+3 − 21 − (|x|−ct) D GD (x, t) ≤ Cα t − 2 5t B 3 (|x|, t) + (1 + t) e . x 2
(4.2)
√ α |x|2 (|x|−ct)2 1 − |α|+3 − − − 5t . Lemma 4.2. For |x| ≥ ct + t, Dx GD (x, t) ≤ Cα t 2 (1+t) 2 e 5t +e Proof. First we let f (x, t) =
1
2
e 0
− (|x|−ctr) 4t
−e
2
− (|x|+ctr) 4t
dr, then
1 − (|x|+ctr)2 r=1 1 − (|x|−ctr)2 r=1 4t 4t + e e ct ct r=0 r=0
(|x|−ct)2 (|x|+ct)2 |x|2 xk 1 =− e− 4t + e− 4t − 2e− 4t . |x| ct
xk ∂ f (x, t) = − ∂xk |x|
Therefore, for |α| ≥ 1, we have
α D f (x, t) = D α−ek xk 1 x x |x| ct −1 ≤ Ct
2 2 2 − (|x|−ct) − (|x|+ct) − |x| 4t 4t 4t +e − 2e e
α xk α (|x|−ct)2 (|x|+ct)2 |x|2 D 1 D 2 e− 4t + e− 4t − 2e− 4t x |x| x α1 +α2 =α−ek
|α|+1 (|x|−ct)2 |x|2 ≤ Cα t − 2 e− 5t + e− 5t .
We have used the fact that in the above√summation |α1 | + |α2 | = |α| − 1 and |x|−|α1 | t −|α2 |/2 ≤ Ct −(|α|−1)/2 if |x| ≥ ct + t. This is because when t ≥ 1, we have
600
D.L. Li
√ |x| ≥ Ct and |x|−|α1 | ≤ Ct −|α1 | ≤ Ct −|α1 |/2 ; when t < 1, we have |x| ≥ C t and |x|−|α1 | ≤ Ct −|α1 |/2 . Next, notice that
2
1 − (|x|−ctr) 4t 0 e
dr ≤ e−
(|x|−ct)2 4t
when |x| ≥ ct,
2 − (|x|−ct) 4t
which implies that |f (x, t)| ≤ Ce . From Lemma 4.1, α D GD (x, t) x
|α 2 |≥1 2 1 ∂ 1 1 α2 |f (x, t)| + ≤ Ct 2 Dxα Dx f (x, t) Dxα1 ∂xj ∂xk |x| |x| α1 +α2 =α+ej +ek |α 2 |≥1 |α2 |+1 (|x|−ct)2 1 ≤ Cα t 2 |x|−(|α|+3) e− 4t + |x|−(|α1 |+1) t − 2 α1 +α2 =α+ej +ek
(|x|−ct)2 |x|2 × e− 5t + e− 5t . When t ≥ 1, |x| ≤ Ct and
α (|x|−ct)2 D GD (x, t) ≤ Cα t 21 t −(|α|+3) e− 4t + x
|α 2 |≥1
t −(|α1 |+1) t −
|α2 |+1 2
α1 +α2 =α+ej +ek
× e ≤ Cα t
− |α|+3 2
2 − (|x|−ct) 5t
(1 + t)
− 21
+e
2 − |x| 5t
2
e
− (|x|−ct) 5t
+e
2
− |x| 5t
.
√ When t < 1, we have |x| ≤ C t, t 1/2 ≤ C(1 + t)−1/2 and 1 ≤ C(1 + t)−1/2 . Therefore
2 2 α |α|+3 (|x|−ct)2 |α|+3 − (|x|−ct) − |x| D GD (x, t) ≤ Cα t − 2 t 21 e− 4t + Cα t − 2 5t 5t e + e x
|α|+3 (|x|−ct)2 |x|2 1 ≤ Cα t − 2 (1 + t)− 2 e− 5t + e− 5t . 5. Pointwise Estimates for GR (x, t) In this section, we shall estimate the remainder GR = G − GD . Part of GR can be computed directly. Define: 0 0 0 T ˆ e = 0 e−|ξ |2 t I − ξ ξ cos(c|ξ |t)(e−|ξ |2 t − ea¯ 2 |ξ |2 t ) 0 G 2 |ξ | 0 0 0 0 0 0 ˆ e,1 − wˆ t G ˆ e,2 0 , = 0 G (5.1) 0 0 0
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ˆ e,1 = e−|ξ |2 t I and G ˆ e,2 = ξ ξ 2 (e−|ξ |2 t − ea¯ 2 |ξ |2 t ). Now we further decompose where G |ξ | ˆ R into two parts G ˆR = G ˆe +G ˆ r . First we notice that G T
|x|2
3
Ge,1 = Ct − 2 e 4t I and therefore by Lemma 3.9, α |α|+3 |x|2 D Ge,1 (x, t) ≤ Cα t − 2 e− 5t .
(5.2)
x
|α|+3 b|x|2 Lemma 5.1. There exists b > 0 such that Dxα Ge,2 (x, t) ≤ Cα t − 2 e− t . Proof. Note that ˆ j k (ξ, t) = G e,2
ξj ξk −|ξ |2 t a¯ 2 |ξ |2 t ξj ξk (e −e )= 2 |ξ | |ξ |2
t
∂ −|ξ |2 s ds = ξj ξk e ∂s
−a¯ 2 t
t −a¯ 2 t
e−|ξ | s ds. 2
Taking inverse Fourier transform, we get jk
Ge,2 (x, t) = C
∂2 ∂xj ∂xk
t
3
−a¯ 2 t
s − 2 e−
|x|2 4s
ds.
Thus, jk
Dxα Ge,2 (x, t) = C
t
3
−a¯ 2 t
s− 2
|x|2 ∂2 Dxα e− 4s ds. ∂xj ∂xk
(5.3)
Let ¯ = max{−a¯ 2 , } and ˜ = min{−a¯ 2 , }. From (5.3) and Lemma 3.9, we have α D Ge,2 (x, t) ≤ Cα x
t −a¯ 2 t
≤ Cα (˜ t)− ≤ Cα t −
s−
|α|+3+2 2
|α|+3+2 2
|α|+3 2
e−
|x|2 5s
ds
|x|2
e− 5¯ t · | + a¯ 2 |t
|x|2
e− 5¯ t .
By the above lemma and Lemma 3.8, we get α |α|+3 D (wt ∗ Ge,2 (x, t)) ≤ Cα,N t − 2 (1 + t)− 21 BN (|x| − ct, t). x
(5.4)
α |α|+3 D Ge (x, t) ≤ Cα,N t − 2 (BN (|x|, t) + (1 + t)− 21 BN (|x| − ct, t)). x
(5.5)
Therefore
Estimating Gr = GR − Ge requires a different approach. In the next few sections, we shall apply Lemma 3.7 to the low, middle and high frequency waves of Gr .
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D.L. Li
5.1. When |ξ | is small. In this section we shall give a pointwise estimate for χ1 (D)Gr . The following proposition follows directly from Theorem 2.1. Proposition 5.2. For sufficiently small |ξ |, ∞
a. g = −
5 bj |ξ |2j ; + 3β j =1
∞
∞
+ 1 b¯2j |ξ |2j ∓ i |ξ | + b¯2j −1 |ξ |2j −1 ; |ξ |2 + 2c2 c j =2 j =2
∞ 3 c. γ = cj |ξ |2j ; κ − ν |ξ |2 + 5 j =2 ∞ ∞ 5ν − 2κ 2 5 d. γ± = |ξ | + c¯2j |ξ |2j ∓ i |ξ | + c¯2j −1 |ξ |2j −1 ; 6c2 3c j =2 j =2 ∞ 50 e. = −i dj |ξ |2j −1 . |ξ | + 9κc b. g± = −
1 5κ
1 2ν
j =2
Corollary 5.3. For sufficiently small |ξ |, # ∞ ∞ ∞ 3κ 1 3 D1 1 2j 1 2j −1 = + Cj |ξ | and D2j |ξ |2j ± i D2j , a. C1 = − κ + −1 |ξ | E1 5 10 j =1 j =1 j =1 # ∞ ∞ ∞ 1 2 D 2 2j −1 b. C2 = Cj2 |ξ |2j and 2 = − + D2j |ξ |2j ± i D2j , −1 |ξ | E2 2 j =1
c. C3 =
9κ + 25 D3 E3
#
∞
j =1
j =1
Cj3 |ξ |2j and
j =1
∞ ∞ 9κ 3 3c 1 3 2j −1 . =− D2j |ξ |2j ± i − · D2j + + −1 |ξ | 50 10 |ξ | j =1
j =1
Remark. From Lemma 3.3 (c), we can see that when m is odd, the higher derivatives (when |β| > m) of |ξ |m behave rather badly for small |ξ |, while even powers of |ξ | tend to have much better derivatives. This is why we cannot directly estimate each entry of ˆ ∗ and G ˆ ± because, as seen in Proposition 5.2 and Corollary 5.3, many coefficients G contain odd powers of |ξ |. Define τ± = λ± Reτ ± iImτ . By we know that Reτ = ∓ ic|ξ | = 2j ∞Theorem 2.12j(b), −1 . The following lemma −( 15 κ + 21 ν)|ξ |2 + ∞ a ¯ |ξ | and Imτ = a ¯ |ξ | j =2 2j j =2 2j −1 ij
is key to estimate the perimeter entries of Gr , i.e., Gr when i = 1, 5 or j = 1, 5. Lemma 5.4. For sufficiently small |ξ | and any multi-indices α and β, ∞ 2β 2j −1 Reτ t D χ1 (ξ )ξ α a. sin(Imτ t)e |ξ | 2j −1 ξ dξ R3 j =l ≤ Cα,β t −
|α|+3+2l−1 2
t |β| , where l ≥ −1,
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∞ 2β α 2j Reτ t − |α|+3+2m 2 χ1 (ξ )ξ cos(Imτ t)e b. 2j |ξ | t |β| , Dξ dξ ≤ Cα,β t 3 R j =m where m≥ 0, ∞ 2β 2j Reτ t D χ1 (ξ )ξ α c. (1 − cos(Imτ t))e |ξ | 2j ξ dξ R3 j =p
|α|+3+2p
|β| ≤ Cα,β t − 2 t , where p ≥−3, ∞ 2β |α|+3+2q 2j a¯ 2 |ξ |2 t Reτ t D χ1 (ξ )ξ α d. (e |ξ | − e ) dξ ≤ Cα,β t − 2 t |β| , 2j ξ R3 j =q 1 1 where q ≥ −2 and a¯ 2 = −( κ + ν). 5 2
Proof. We shall only present the proof for (5.4). 3 ∞ ∞ 1 1 sin(Imτ t) = a¯ 2j −1 |ξ |2j −1 t − a¯ 2j −1 |ξ |2j −1 t 3 + · · · + F (Imτ t) 3! |β|! j =2 j =2 ∞ 1 = Aj,k |ξ |2j −1 t 2k−1 + F (Imτ t), (|β|)! |β|+1 1≤k≤
j =3k−1
2
where F (x) = 0 sin(|β|+1) (s)(x − s)|β| ds. Therefore, ∞ 2j −1 |ξ |2j −1 sin(Imτ t) x
j =l
=
1≤k≤ |β|+1 2
∞ j =l+3k−2
∞ 1 2j,k |ξ |2j t 2k−1 + 2j −1 |ξ |2j −1 F (Imτ t) |β|! j =l
= I + I I. Since l ≥ −1, j = l +3k −2 ≥ 0. Note that the lemma is not true without the hypothesis β l ≥ −1 since when j < 0, Dξ |ξ |2j can be arbitrarily large when |ξ | is small. By Lemma 3.3 (c), we have ∞ β 2j 2k−1 t 2j,k |ξ | Dξ j =l+3k−2 1≤k≤ |β| 2 ≤ Cβ t 2k−1 + |ξ |2(l+3k−2)−|β| t 2k−1 . (5.6) 1≤k≤ |β|−2l+4 6
|β|−2l+4
For II, we first notice that by Lemma 3.14, we have for all β2 ≤ β, |Dξ 2 F (Imτ t)| ≤ Cβ |ξ |3|β|+3−|β2 | t |β|+1 ≤ Cβ |ξ |3|β|+2−|β2 | t |β|+1 . β
(5.7)
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Hence, β Dξ (I I ) ≤ Cβ
∞ β1 2j −1 β2 D |ξ | F (Imτ t) D 2j −1 ξ ξ β1 +β2 =β j =l
≤ Cβ |ξ |2|β|+2+2l−1 t |β|+1 .
(5.8)
By (5.6) and (5.8), ∞ β 2j −1 D sin(Imτ t) |ξ | 2j −1 ξ j =l t 2k−1 + ≤ Cβ 1≤k≤ |β|−2l+4 6
|ξ |2(l+3k−2)−|β| t 2k−1 + |ξ |2|β|+2+2l−1 t |β|+1 .
|β|−2l+4
(5.9)
Now, by Lemma 3.4 and (5.9), ∞ 2β 2j −1 Reτ t D ξ α sin(Imτ t)e |ξ | 2j −1 ξ j =l β ∞ 1 ≤α β1 α β2 Reτ t β3 2j −1 ≤ sin(Imτ t) 2j −1 |ξ | Dξ (ξ ) Dξ e Dξ β1 +β2 +β3 =2β j =l ≤ Cβ
β 1 ≤α β1 +β2 +β3 =2β
|ξ ||α|−|β1 |
t 2k −1 +
|β |−2l+4 1≤k ≤ 3 6
tk +
|β | 1≤k< 22
·
|β3 |−2l+4 |β |+1
|β2 | 2 ≤k≤|β2 |
Reτ t |ξ |2k−|β2 | t k e
|ξ |2(l+3k −2)−|β3 | t 2k −1 +|ξ |2|β3 |+2+2l−1 t |β3 |+1 .
Note that each of the above terms can be written as |ξ |p t q eReτ t , where p ≥ −1. Thus we can apply Lemma 3.5 to estimate each of the above terms. For instance, we look at the term involving |ξ |2|β3 |+2+2l−1 t |β3 |+1 with |β22 | ≤ k ≤ |β2 |, χ1 (ξ )|ξ ||α|−|β1 |+2k−|β2 |+2|β3 |+2+2l−1 t k+|β3 |+1 eReτ t dξ R3
≤ Cα,β t −
|α|−|β1 |+2k−|β2 |+2|β3 |+2+2l−1+3 +k+|β3 |+1 2
≤ Cα,β t −
|α|+n+2l−1 2
t |β| .
The other terms can be estimated similarly. Hence, ∞ 2β |α|+n+2l−1 2j −1 Reτ t D χ1 (ξ )ξ α 2 sin(Imτ t)e dξ ≤ Cα,β t − |ξ | t |β| . 2j −1 ξ n R j =l
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By Lemma 3.7 and the above lemma, we have the following corollary. Corollary 5.5. Let α be a multi-index, ξ, x ∈ R3 and t ≥ 0. ∞ 2j −1 |ξ |2j −1 sin(Imτ t)eReτ t , a. Suppose that for sufficiently small |ξ |, fˆ(ξ, t) = j =l
|α|+3+2l−1
2 ≤ Cα,N t − BN (|x|, t) for any ∞ b. Suppose that for sufficiently small |ξ |, fˆ(ξ, t) = 2j |ξ |2j cos(Imτ t)eReτ t ,
where l ≥ −1. Then N ∈ N.
|Dxα (χ1 (D)f (x, t))|
j =m
where m ≥
− |α|+3+2m 2
0. Then |Dxα (χ1 (D)f (x, t))|
≤ Cα,N BN (|x|, t t) for any N ∈ N. ∞ c. Suppose that for sufficiently small |ξ |, fˆ(ξ, t) = 2j |ξ |2j (1−cos(Imτ t))eReτ t , j =p
|α|+3+2p
≤ Cα,N t − 2 BN (|x|, t) for any ∞ 2 d. Suppose that for sufficiently small |ξ |, fˆ(ξ, t) = 2j |ξ |2j (ea¯ 2 |ξ | t − eReτ t ), where p ≥ −3. Then N ∈ N.
|Dxα (χ1 (D)f (x, t))|
where q ≥ −2. Then N ∈ N.
|Dxα (χ1 (D)f (x, t))|
j =q
≤ Cα,N t −
Lemma 5.6. Suppose that α and β are multi-indices, and
|α|+3+2q 2
∞
BN (|x|, t) for any
2j |ξ |2j is convergent
j =m
for sufficiently small |ξ |, where m ≥ 0. Then ∞ 2β |α|+3+2m 2j λt D χ1 (ξ )ξ α e dξ ≤ Cα,β t − 2 t |β| . 2j |ξ | ξ R3 j =m Proof. First we have ∞ 2β 2j λt D χ1 (ξ )ξ α e 2j |ξ | ξ j =m ≤ Cα,β
β1 ≤α,|β 2 |≤2m
χ1 (ξ )|ξ ||α|−|β1 |+2m−|β2 |
β1 +β2 +β3 ≤2β
× 1≤k<
+Cα,β
|β3 | 2
tk +
|β3 | 2 ≤k≤|β3 |
β1 ≤α,|β 2 |>2m β1 +β2 +β3≤2β
|ξ |2k−|β3 | t k eλt
χ1 (ξ )|ξ ||α|−|β1 | 1≤k<
|β3 | 2
tk +
|β3 | 2 ≤k≤|β3 |
|ξ |2k−|β3 | t k eλt .
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We have used Lemma 3.4. We can now complete the proof by using Lemma 3.5 to estimate each of the above terms in the same way as in Lemma 5.4. ∞ 2j eλt , Corollary 5.7. Suppose that for sufficiently small |ξ |, fˆ(ξ, t) = |ξ | 2j j =m α − |α|+3+2m 2 where m ≥ 0. Then Dx (χ1 (D)f (x, t)) ≤ Cα,N t BN (|x|, t) for any N ∈ N. ij
ij
Note that for i = 1, 5 or j = 1, 5, Gd = 0 and therefore Gr = Gij . Write ∗ ij ˆ =G ˆ + eλ+ t + G ˆ − eλ− t , then Gij G r = G = Gij + Gij . We have the following proposition on the perimeter entries of Gr . Proposition 5.8. For i = 1, 5 or j = 1, 5, |α|+3 a. Dxα (χ1 (D)G∗ij (x, t)) ≤ Cα,N t − 2 BN (|x|, t). |α|+3 1 b. Dxα (χ1 (D)Gij (x, t)) ≤ Cα,N t − 2 (1 + t)− 2 BN (|x| − ct, t). Proof. (a) follows directly from Proposition 5.2, Corollary 5.3 and Corollary 5.7. For (b), we note that: ˆ = G ˆ + eτ+ t (wˆ t + ic|ξ |w) ˆ − eτ− t (wˆ t − ic|ξ |w) G ˆ +G ˆ ij ij ij = wˆ t (fˆij1 + fˆij2 ) + w( ˆ fˆij3 + fˆij4 ), where ˆ+ +G ˆ − ) cos(Imτ t)eReτ t , ˆ+ −G ˆ − ) sin(Imτ t)eReτ t , fˆij2 = i(G fˆij1 = (G ij ij ij ij ˆ+ −G ˆ − ) cos(Imτ t)eReτ t , fˆ4 = −c|ξ |(G ˆ+ +G ˆ − ) sin(Imτ t)eReτ t . fˆij3 = ic|ξ |(G ij ij ij ij ij ˆ+ +G ˆ −: By Proposition 5.2 and Corollary 5.3, we have for G
iξ T T 2 g+ D1 + g− E1 = O(|ξ | ), (g+ D2 + g− E2 ) · |ξ |2 = iξ O(1), iξ(D1 + E1 ) = iξ O(1), iξ T γ+ D1 + γ− E1 = O(|ξ |2 ), (γ+ D2 + γ− E2 ) · 2 = iξ T O(1), |ξ |
g+ D3 + g− E3 = O(1) iξ(D3 + E3 ) = iξ O(1) , γ+ D3 + γ− E3 = O(1)
ˆ+ −G ˆ −: and for G
iξ T 1 T g+ D1 − g− E1 = iO(|ξ |), (g+ D2 − g− E2 ) · |ξ |2 = ξ O |ξ | , g+ D3 − g− E3 = iO(|ξ |)
1 iξ(D1 − E1 ) = ξ O(|ξ |), , iξ(D3 − E3 ) = ξ O . |ξ |
iξ T 1 , γ+ D3 − γ− E3 = iO(|ξ |) γ+ D1 − γ− E1 = iO(|ξ |), (γ+ D2 − γ− E2 ) · 2 = ξ T O |ξ | |ξ |
By Corollary 5.5, we have |Dxα (χ1 (D)fijk (x, t))| ≤ Cα,N t − |Dxα (χ1 (D)fijk (x, t))| ≤ Cα,N t
|α|+3 2
− |α|+4 2
BN (|x|, t) for k = 1, 2, BN (|x|, t) for k = 3, 4.
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Therefore, by Lemma 3.8 and the above estimates, we have |α|+3 1 α Dx (wt ∗ χ1 (D)fijk (x, t)) ≤ Cα,N t − 2 (1 + t)− 2 BN (|x| − ct, t) for k = 1, 2, |α|+3 1 α Dx (w ∗ χ1 (D)fijk (x, t)) ≤ Cα,N t − 2 (1 + t)− 2 BN (|x| − ct, t) for k = 3, 4. Therefore, α Dx (χ1 (D)Gij (x, t)) = Dxα wt ∗ χ1 (D)(fij1 + fij2 ) + w ∗ χ1 (D)(fij3 + fij4 ) ≤ Cα,N t −
|α|+3 2
1
(1 + t)− 2 BN (|x| − ct, t).
ˆ r , i.e., G ˆ ij We now turn our attention to the central block of G r , 2 ≤ i, j ≤ 4: T λt λ+ t λ− t a¯ 2 |ξ |2 t ˆ cr = − ξ ξ G C e + D e + E e + w ˆ e 2 2 2 t |ξ |2 ξ ξ T ˆ c,+ c,− a¯ 2 |ξ |2 t ˆ c,∗ ˆ =G − + G + w ˆ e G , t r r r |ξ |2 where ˆ c,∗ G r ˆ c,± G r
∞ ξ ξ T 2 2j λt e , =− 2 Cj |ξ | |ξ | j =1 ∞ ∞ 1 2 2 2j −1 λ± t e . = − + D2j |ξ |2j ± i D2j −1 |ξ | 2 j =1
Since
ˆ c,∗ G r
j =1
contains only non-negative even powers of |ξ |, we have by Corollary 5.7: α c,∗ |α|+3 D G (x, t) ≤ Cα,N t − 2 BN (|x|, t). (5.11) x r
ˆ c,± For G r , notice that ˆ c,± G r
∞ ∞ 1 2 2 2j −1 λ± t e = − + D2j |ξ |2j ± i D2j −1 |ξ | 2 j =1 j =1 ∞ ∞ 1 2 2 2j −1 = − + D2j |ξ |2j ± i D2j −1 |ξ | 2
j =1
×e Therefore,
(5.10)
Reτ t
j =1
(cos(Imτ t) ± sin(Imτ t))(wˆ t ± ic|ξ |w). ˆ
ˆ c,+ ˆ c,− G +G = wˆ t eReτ t −1 + 2 r r
∞ j =1
2 D2j |ξ |2j cos(Imτ t)
∞ 2 2j −1 −2 sin(Imτ t) D2j −1 |ξ | j =1
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∞ 2 2j −1 cos(Imτ t) −c|ξ |we ˆ Reτ t 2 D2j −1 |ξ | j =1
+ −1 + 2
∞
2 D2j |ξ |2j sin(Imτ t)
j =1
and ξξT ξ ξ T ˆ c,+ Reτ t ˆ c,− ( G + G ) = − w ˆ e cos(Imτ t) + wˆ t Rˆ 1 + wˆ Rˆ 2 , t r r |ξ |2 |ξ |2 where
Rˆ 1 = ξ ξ T eReτ t 2
∞ j =0
and
2 2j cos(Imτ t) − 2 D2j +2 |ξ |
∞ j =0
(5.12)
2 2j −1 sin(Imτ t) D2j +1 |ξ |
∞ 2 2j cos(Imτ t) D2j Rˆ 2 = −cξ ξ T eReτ t 2 +1 |ξ | j =0
∞ 1 2 + − D2j |ξ |2j −1 sin(Imτ t) . +2 |ξ | j =1
By Corollary 5.5, we have α |α|+3 D (χ1 (D)R1 (x, t)) ≤ Cα,N t − 2 BN (|x|, t) x and
α |α|+4 D (χ1 (D)R2 (x, t)) ≤ Cα,N t − 2 BN (|x|, t), x
which implies α |α|+3 D (wt ∗ χ1 (D)R1 (x, t)) ≤ Cα,N t − 2 (1 + t)− 21 BN (|x| − ct, t), x α |α|+3 D (w ∗ χ1 (D)R2 (x, t)) ≤ Cα,N t − 2 (1 + t)− 21 BN (|x| − ct, t). x
(5.13) (5.14)
ˆ cr as follows: From (5.10) and (5.12), we can rewrite G T
ξξ 2 ˆ c,∗ ˆ cr = G ˆ t Rˆ 1 − wˆ Rˆ 2 − wˆ t 2 (ea¯ 2 |ξ | t − eReτ t cos(Imτ t)). G r −w |ξ |
(5.15)
It is clear that we now have to estimate the last term on the right-hand side of (5.15). Recall that by our definition, a¯ 2 = −( 15 κ + 21 ν) is the leading coefficient of Reτ , wˆ t
ξ ξ T a¯ 2 |ξ |2 t (e − eReτ t cos(Imτ t)) |ξ |2 % ξξT $ 2 = wˆ t 2 (ea¯ 2 |ξ | t − eReτ t ) + eReτ t (1 − cos(Imτ t)) |ξ | = wˆ t (Tˆ1 + Tˆ2 ).
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By Corollary 5.5 and Lemma 3.8, α |α|+3 D (wt ∗ χ1 (D)Tj (x, t)) ≤ Cα,N t − 2 (1 + t)− 21 BN (|x| − ct, t) for j = 1, 2. x (5.16) From (5.11)–(5.16), we have proved the following proposition. Proposition 5.9. For 2 ≤ i, j ≤ 4, |α|+3 1 α ij Dx (χ1 (D)Gr (x, t)) ≤ Cα,N t − 2 (BN (|x|, t) + (1 + t)− 2 BN (|x| − ct, t)). Finally we summarize this section in the following proposition. |α|+3 1 Proposition 5.10. Dxα (χ1 (D)Gr (x, t)) ≤ Cα,N t − 2 (BN (|x|, t)+(1+t)− 2 BN (|x|− ct, t)). 5.2. When |ξ | is in the middle. In this section we shall estimate χ2 (D)Gr . Note that ˆr = G ˆ λ eλt + G ˆ + eλ+ t + G ˆ − eλ− t − G ˆ a¯ 2 ea¯ 2 |ξ |2 t = G ˆ r,1 − G ˆ r,2 . G where
iξ T gC1 gC2 |ξ |2 gC3 ξξT ˆ Gλ = iξ C1 −C2 iξ C3 , 2 |ξ | iξ T γ C 1 γ C2 2 γ C 3 |ξ |
0
0
0
(5.17)
T ˆ a¯ 2 = 0 ξ ξ wˆ t 0 . G |ξ |2 0 0 0
ˆ a¯ 2 ∈ C ∞ (U ), where U = {ξ : ι ≤ |ξ | ≤ R + 1}, we have |D β G ˆ Since G ξ a¯ 2 | ≤ Cβ and |Dξ ea¯ 2 |ξ | t | ≤ Cβ e−bt in U for any multi-index β. Hence 2β α ˆ a¯ 2 ea¯ 2 |ξ |2 t dξ ≤ Cα,β e−bt . Dξ ξ χ2 (ξ )G β
2
R3
By Lemma 3.7, therefore, we have α |α|+3 D (χ2 (D)Gr,2 (x, t)) ≤ Cα,N t − 2 BN (|x|, t). x
(5.18)
ˆ r,1 , the concern is that, by Cardano’s formula, λ and λ± contain cube roots and For G square roots and therefore are not differentiable at certain points. Also, vanishes when the characteristic polynomial (2.3) has repeated roots and it appears in the denominator ˆ r,1 is not differentiable at these of Cj , Dj , and Ej . This, however, is not to say that G ˆ points. Note that Gr,1 is a symmetric expression in λ and λ± , i.e., any permutation of the ˆ r,1 unchanged. As we shall prove in this section, G ˆ r,1 is actually three roots will leave G analytic in ξ despite the fact that λ and λ± are not. Let
10 2 3 2 2 ˜ ˜ ˜ () f (z, λ) = λ + (κ + ν)zλ + νκz + z λ˜ + κz2 , z, λ˜ ∈ C. 9 3 ˜ |2 ) and λ± (ξ ) = Let λ˜ (z) and λ˜ ± (z) be the three roots of () = 0. Then λ(ξ ) = λ(|ξ 2 ˜ correspondingly as in (2.2) and (2.6). ˜ g˜ ± , γ˜ , γ˜± , C˜ j , D˜ j , C˜ j and λ˜ ± (|ξ | ). Define g,
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˜ Lemma 5.11. (z) = 0 if and only if () = 0 has repeated roots. ˜ = −γ˜− g˜ + +γ˜− g+ ˜ γ˜ g˜ + +γ˜+ g˜ − −γ˜+ g− ˜ γ˜ g˜ − , Proof. Substituting g, ˜ g˜ ± , γ˜ and γ˜± into we get ˜ =
z (λ˜ + + κz)(λ˜ − + κz)(λ˜ + κz)
˜ (λ˜ − λ˜ + )(λ˜ + − λ˜ − )(λ˜ − − λ).
Lemma 5.12. The following functions are analytic when |z| ≥ δ. ˜ ˜ ˜ a. g(z) ˜ C˜ j (z)eλ(z)t + g˜ + (z)D˜ j (z)eλ+ (z)t + g˜ − (z)E˜ j (z)eλ− (z)t , ˜ ˜ ˜ b. C˜ j (z)eλ(z)t + D˜ j (z)eλ+ (z)t + E˜ j (z)eλ− (z)t , ˜ ˜ ˜ c. γ˜ (z)C˜ j (z)eλ(z)t + γ˜+ (z)D˜ j (z)eλ+ (z)t + γ˜− (z)E˜ j (z)eλ− (z)t . ˜ ˜ ˜ Proof. We shall only present the proof for C˜ 3 (z)eλ(z)t + D˜ 3 (z)eλ+ (z)t + E˜ 3 (z)eλ− (z)t . The proofs for the other functions in the lemma are analogous. ˜ ˜ ˜ C˜ 3 (z)eλ(z)t + D˜ 3 (z)eλ+ (z)t + E˜ 3 (z)eλ− (z)t ˜
=
=
˜
˜
λ+ (z)t + (g(z) ˜ ˜ − g˜ + (z))eλ− (z)t (g˜ + (z) − g˜ − (z))eλ(z)t + (g˜ − (z) − g(z))e z ˜ (λ˜ − λ˜ + )(λ˜ + − λ˜ − )(λ˜ − − λ) ˜ ˜ (λ+ + κz)(λ− + κz)(λ˜ + κz)
˜ ˜ λ˜ + , λ˜ − , t) λ, (λ˜ + + κz)(λ˜ − + κz)(λ˜ + κz)(λ˜ − λ˜ + )(λ˜ + − λ˜ − )(λ˜ − − λ)A( , ˜ 2 λ˜ λ˜ + λ˜ − (λ˜ − λ˜ + )2 (λ˜ + − λ˜ − )2 (λ˜ − − λ) (5.19)
where ˜ λ˜ + , λ˜ − , t) A(λ, ˜ ˜ ˜ λ˜ − (z)t = (λ˜ λ˜ − − λ˜ λ˜ + )eλ(z)t + (λ˜ + λ˜ − λ˜ + λ˜ − )eλ+ (z)t + (λ˜ − λ˜ + − λ˜ − λ)e ∞ ∞ m ˜ (λ(z)t) (λ˜ + (z)t)m = (λ˜ λ˜ − − λ˜ λ˜ + ) + (λ˜ + λ˜ − λ˜ + λ˜ − ) m! m! m=1
m=1
∞ (λ˜ − (z)t)m ˜ +(λ˜ − λ˜ + − λ˜ − λ) . m! m=1
Note that the numerator is a symmetric power series in λ˜ and λ˜ ± . It is well known that every symmetric polynomial can be written as a power sum of the elementary symmetric polynomials λ˜ + λ˜ + + λ˜ − , λ˜ λ˜ + + λ˜ λ˜ − + λ˜ + λ˜ − and λ˜ λ˜ + λ˜ − . Since λ˜ + λ˜ + + λ˜ − = −(κ + ν)z,
λ˜ λ˜ + + λ˜ λ˜ − + λ˜ + λ˜ − = νκz2 +
10 z, 9
2 λ˜ λ˜ + λ˜ − = − κz2 , 3 we know that the numerator can be written as a power series in z and therefore is entire in z. Similarly, we notice that the denominator is a symmetric polynomial in λ˜ and λ˜ ± ,
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˜ ˜ so it can be written as a polynomial in z. This proves that C˜ 3 (z)eλ(z)t + D˜ 3 (z)eλ+ (z)t + ˜ E˜ 3 (z)eλ− (z)t is a meromorphic function. The possible poles occur when either (∗) = 0 has repeated roots or one of λ˜ and λ˜ ± is 0. Observe that one of λ˜ and λ˜ ± is 0 if and only if z = 0. Thus, for |z| ≥ δ, the poles occur only when (∗) = 0 has repeated roots. We first point out that (∗) = 0 has no roots of multiplicity 3 except at z = 0. This is because if λ˜ is a root of multiplicity 3, then λ˜ 3 = −2κz2 /3 and 3λ˜ = −(κ + ν)z, which gives (κ + ν)3 z3 = 18κz2 ⇒ z = 0 or z = 18κ/(κ + ν)3 . But z = 18κ/(κ + ν)3 does not satisfy 3λ˜ 3 = νκz2 + 10z/9. Therefore,(∗) = 0 has a root of multiplicity 3 only when z = 0. Now, without loss of generality, assume that λ˜ + (z∗ ) = λ˜ − (z∗ ) for some z∗ . Note that there are only finitely many such z∗ since the denominator in (5.19) can be written as a polynomial in z and therefore can only have finitely many roots. Let B denote a ˜ neighborhood of z∗ such that there are no other repeated roots. Note that λ(z) is bounded ˜ away from λ± (z) in B because there cannot be a root of multiplicity 3. Thus,
1 1 z − g˜ + − g˜ − λ˜ + λ˜ − C˜ 3 = = z ˜ ˜ (λ˜ − λ˜ + )(λ˜ + − λ˜ − )(λ˜ − − λ) (λ˜ + + κz)(λ˜ − + κz)(λ˜ + κz) (λ˜ + + κz)(λ˜ − + κz)(λ˜ + κz) = λ˜ + λ˜ − (λ˜ − λ˜ + )(λ˜ − λ˜ − ) ˜
is bounded in B. Therefore C˜ 3 (z)eλ(z)t is bounded in B. Next, we notice that ˜
˜
λ+ (z)t + (g(z) (g˜ − (z) − g(z))e ˜ ˜ − g˜ + (z))eλ− (z)t λ˜ + − λ˜ − ˜
˜
˜
˜ λ+ (z)t − eλ− (z)t ) (g˜ − − g˜ + )eλ− (z)t + (g˜ − − g)(e λ˜ + − λ˜ − λ˜ + 1 ˜ = zeλ− (z)t + (g˜ − − g) ˜ est ds. λ˜ + − λ˜ − λ˜ − =
The first term of the above equality is bounded in B and the second term converges ˜ to (g˜ − (z∗ ) − g(z ˜ ∗ ))eλ− (z∗ )t as z → z∗ and therefore is also bounded in B. Hence, ˜ ˜ D˜ 3 (z)eλ+ (z)t + E˜ 3 (z)eλ− (z)t is bounded in B because ˜ ˜ D˜ 3 (z)eλ+ (z)t + E˜ 3 (z)eλ− (z)t
=−
$ % ˜ λ˜ + (z)t + (g(z) ˜ (λ˜ + + βz)(λ˜ − + βz)(λ+βz) (g˜ − (z)− g(z))e ˜ ˜ − g˜ + (z))eλ− (z)t ˜ z(λ˜ − λ˜ + )(λ˜ + − λ˜ − )(λ˜ − − λ) ˜
=−
˜
λ+ (z)t + (g(z) ˜ ˜ ˜ − g˜ + (z))eλ− (z)t (g˜ − (z) − g(z))e (λ˜ + +βz)(λ˜ − + βz)(λ+βz) · . ˜ λ˜ + )(λ˜ − − λ) ˜ z(λ− λ˜ + − λ˜ −
˜ ˜ ˜ Since C˜ 3 (z)eλ(z)t + D˜ 3 (z)eλ+ (z)t + E˜ 3 (z)eλ− (z)t is bounded at the potential poles except at z = 0, it must be analytic when |z| ≥ δ.
612
D.L. Li
|α|+3 Lemma 5.13. Dxα (χ2 (D)Gr,1 (x, t)) ≤ Cα,N t − 2 BN (|x|, t). Proof. Note that
R3
e
ix·ξ
=e
2β Dξ
−bt
≤ e−bt .
R3
ˆ ξ χ2 (ξ )Gr,1 (ξ, t) dξ
α
e
ix·ξ
2β Dξ
ˆ λ e(λ+b)t + G ˆ + e(λ+ +b)t + G ˆ − e(λ− +b)t ξ χ2 (ξ ) G α
dξ
We have used Theorem 2.2 and Lemma 5.12 in the last step. The lemma then follows directly from Lemma 3.7. By (5.18) and Lemma 5.13, we have proved the following proposition. |α|+3 Proposition 5.14. Dxα (χ2 (D)Gr (x, t)) ≤ Cα,N t − 2 BN (|x|, t). 5.3. When |ξ | is large. In this section we shall estimate χ3 (D)Gr . The following proposition follows directly from Theorem 2.3. Proposition 5.15. For sufficiently large |ξ |, ∞
∞
j =0 ∞
j =1
1 3 bj |ξ |−2j , g+ = − |ξ |2 + bj+ |ξ |−2j , a. g = − ν|ξ |2 + 2 ν 1 g− = − |ξ |2 + β ∞
b. γ = c. =
cj |ξ |
j =−1 ∞
e. C2 =
j =1
−2j
, γ+ =
cj+ |ξ |−2j , γ−
∞
=
cj− |ξ |−2j ;
j =−1
dj |ξ |−2j ; ∞
∞
∞
2 −2 1 −2j Cj |ξ | , D1 = Dj1 |ξ |−2j , E1 = Ej1 |ξ |−2j ; |ξ | + 3ν
∞
j =2
Cj2 |ξ |−2j , D2 =
j =1
f. C3 =
∞ j =0
j =−2
d. C1 = −
bj− |ξ |−2j ;
∞ j =2
∞
j =1
Dj2 |ξ |−2j , E2 =
j =0
Cj3 |ξ |−2j , D3 =
∞ j =1
∞
j =1
Ej2 |ξ |−2j ;
j =1
Dj3 |ξ |−2j , E3 =
∞
Ej3 |ξ |−2j .
j =1
Recall that ˆ λ eλt + G ˆ + eλ+ t + G ˆ − eλ− t − G ˆ a¯ 2 ea¯ 2 |ξ |2 t = G ˆδ +G ˆ −G ˆ r,2 . ˆr = G G
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
613
By Theorem 2.3 and Proposition 5.15, we have 2β α ˆ + eλ+ t dξ ≤ Cα,β e−bt , Dξ ξ χ3 (ξ )G 3 R 2β α ˆ − eλ− t dξ ≤ Cα,β e−bt , Dξ ξ χ3 (ξ )G 3 R 2β α ˆ a¯ 2 ea¯ 2 |ξ |2 t dξ ≤ Cα,β e−bt . Dξ ξ χ3 (ξ )G R3
Therefore, by Lemma 3.7, |Dxα (χ3 (D)G (x, t))| ≤ Cα,N t − |Dxα (χ3 (D)Gr,2 (x, t))| ≤ Cα,N t ˆδ = G ˆλ We now consider G
eλt .
|α|+3 2
− |α|+3 2
BN (|x|, t),
(5.20)
BN (|x|, t).
(5.21)
Note that by Theorem 2.3, ∞
λ=−
2 + aj |ξ |−2j , 3ν j =1
thus
eλt = e
2 − 3ν t
1 +
∞
aj |ξ |−2j t + · · · +
j =1
j =1
=e
2 − 3ν t
1 +
∞
pj (t)|ξ |−2j
m aj |ξ |−2j
j =1
∞ +Rm aj |ξ |−2j t
∞
tm m!
∞ + Rm aj |ξ |−2j t ,
j =1
j =1
x
1 es (x − s)m ds and pj (t)’s are polynomials in t of degree no m! 0 larger than m. From Proposition 5.15, we have where Rm (x) =
gC1 = 1 + γ C1 =
∞
j =1 ∞
5 + 3
gC1 e
−2j 3,1 γ C2 = j |ξ |
j =1
Therefore, λt
−2j 1,1 gC2 = j |ξ |
∞
−2j 1,2 gC3 = j |ξ |
j =0 ∞
−2j 3,2 γ C3 = j |ξ |
j =0
∞ j =1 ∞
−2j 1,3 , j |ξ | −2j 3,3 . j |ξ |
j =1
=e
2 t − 3ν
∞ ∞ ∞ 1,1 1,1 −2j −2j −2j 1 + Rm pj (t)|ξ | + 1 + j |ξ | aj |ξ | t ,
j =1
j =1
j =1
j =1
2 t iξ T λt 1,2 1,2 T − 3ν −2j −2j −2j Rm pj (t)|ξ | + j −1 |ξ | aj |ξ | t , gC2 2 e = iξ e |ξ | j =1 j =1 j =1 ∞ ∞ ∞ 2 t 1,3 1,3 − 3ν λt −2j −2j −2j Rm pj (t)|ξ | + j |ξ | aj |ξ | t , gC3 e = e ∞
j =1
∞
j =1
∞
614
D.L. Li iξ C1 e
λt
= iξ e
2 t − 3ν
∞ ∞ ∞ 2,1 −2j 1 −2j −2j Rm pj (t)|ξ | + Cj |ξ | aj |ξ | t , j =1
j =1
j =2
j =2
j =1
∞ ∞ ∞ 2 2,2 pj (t)|ξ |−2j + Cj2−1 |ξ |−2j Rm aj |ξ |−2j t , −C2 2 eλt = −ξ ξ T e− 3ν t |ξ | j =2 j =2 j =1 ∞ ∞ ∞ 2 t 2,3 − 3ν λt −2j 3 −2j −2j Rm pj (t)|ξ | + Cj |ξ | aj |ξ | t , iξ C1 e = iξ e ξξT
j =1
∞ 5 5 −2j + pj3,1 (t)|ξ |−2j + + 3,1 aj |ξ |−2j t , Rm γ C1 eλt = j |ξ | 3 3 j =1 j =1 j =1 ∞ ∞ ∞ 2 t iξ T λt 3,2 3,2 T − 3ν −2j −2j −2j Rm pj (t)|ξ | + j −1 |ξ | aj |ξ | t , γ C2 2 e = iξ e |ξ | j =1 j =1 j =1 ∞ ∞ ∞ 2 −2j Rm γ C3 eλt = e− 3ν t pj3,3 (t)|ξ |−2j + 3,3 aj |ξ |−2j t . j |ξ |
∞
2 e− 3ν t
j =1
Hence,
∞
∞
j =1
pj1,1 (t)|ξ |−2j
∞
j =1
pj1,2 (t)|ξ |−2j
∞
pj1,3 (t)|ξ |−2j
iξ 1+ j =1 j =1 j =1 ∞ ∞ ∞ 2,1 2,2 2,3 −2j T −2j −2j e− 3ν2 t ˆ δ = iξ p (t)|ξ | −ξ ξ p (t)|ξ | iξ p (t)|ξ | G j j j j =1 j =2 j =2 ∞ ∞ ∞ 5 3,1 3,2 3,3 −2j T −2j −2j + pj (t)|ξ | iξ pj (t)|ξ | pj (t)|ξ | 3 j =1 j =1 j =1 ∞ 2 ˆ λ Rm +G aj |ξ |−2j t e− 3ν t . (5.22) j =1
T
|α|+2
Recall that Fα (x, t) =
Lj δ(x) e−Ct , where Lj ’s are defined in Theorem 1.1.
j =0
We have the following lemma. |α|+3 Lemma 5.16. Dxα (χ3 (D)(Gδ (x, t) − Fα (x, t))) ≤ Cα,N t − 2 BN (|x|, t). Proof. By Lemma 3.7, we only need to prove |α|+3 ix·ξ 2β α p.v. ˆ δ (ξ, t) − Fˆα (ξ, t)) dξ ≤ Cα,β t − 2 t |β| . e Dξ ξ χ3 (ξ )(G R3
(5.23)
We shall only prove the lemma for even |α| since the proof for odd |α| is analogous. In this case, |α| |α| +1 +1 2 2 2 L2j δ(x) + L2j −1 δ(x) e− 3ν t . Fα (x, t) = j =0
j =1
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
615
Taking Fourier transform, we get,
|α| 2 +1
|α| 2 +1
|α| 2 +1
j =1
j =1
j =1
1 + pj1,1 (t)|ξ |−2j iξ T pj1,2 (t)|ξ |−2j pj1,3 (t)|ξ |−2j j =1 j =1 j =1 |α| +1 |α| |α| +2 +1 2 2 2 2,1 2,2 2,3 2 −2j T −2j −2j Fˆα = iξ e− 3ν t . p (t)|ξ | −ξ ξ p (t)|ξ | iξ p (t)|ξ | j j j j =1 j =2 j =2 |α| |α| |α| +1 +1 +1 2 2 2 3,1 3,2 3,3 5 + pj (t)|ξ |−2j iξ T pj (t)|ξ |−2j pj (t)|ξ |−2j 3
ˆ − 3ν2 t + G ˆ λ Rm ˆ δ − Fˆα = Ae From 5.22 and the above, we get G where
∞
pj1,1 (t)|ξ |−2j
T
∞
pj1,2 (t)|ξ |−2j
∞ −2j t j =1 aj |ξ |
∞
2
e− 3ν t ,
pj1,3 (t)|ξ |−2j
iξ |α| |α| j = |α| j = 2 +2 j = 2 +2 2 +2 ∞ ∞ ∞ 2,1 2,2 2,3 −2j T −2j −2j iξ p (t)|ξ | −ξ ξ p (t)|ξ | iξ p (t)|ξ | . ˆ A = j j j |α| |α| j = |α| j = 2 +3 j = 2 +2 2 +2 ∞ ∞ ∞ pj3,1 (t)|ξ |−2j iξ T pj3,2 (t)|ξ |−2j pj3,3 (t)|ξ |−2j j = |α| 2 +2
j = |α| 2 +2
j = |α| 2 +2
|α| 2 e− 3ν t . Let m = + 1 so that the poly2 |α| nomials pjk,l (t) all have degree less than or equal to + 1. Note that for k ≤ m, 2 x 1 (k) (k) Rm (x) = es (x − s)m−k ds, and for k > m, Rm (x) = ex . Therefore, by (m − k)! 0 Lemma 3.2, we have ˆ λ Rm We first estimate G
∞ −2j t j =1 aj |ξ |
∞ β −2j D Rm a |ξ | t j ξ j =1 |β| |η ∞ k l |≥1 (k) −2j k R ≤ a |ξ | t t |ξ |2−|ηl | j m k k=1 j =1 l=1 ηl =β
l=1
≤
∞ (k) −2j k R aj |ξ | t t |ξ |−2k−|β| m j =1
m∧|β| k=1
616
D.L. Li
|β| ∞ (k) −2j k + aj |ξ | t t |ξ |−2k−|β| Rm k=m+1 j =1 m−k+1 m∧|β| ∞ ∞ −2j 1 a |ξ | t −2j ≤ e j =1 j a |ξ | t m−k+1 j (m − k)! k=1 j =1 |β| ×t k |ξ |−2k−|β| + t k |ξ |−2k−|β| ≤ Cβ |ξ |
k=m+1 ∞ −2j t −2m−2−|β| |β|∨(m+1) j =1 aj |ξ |
t
e
.
β ˆ 3 Also, |Dξ G λ | ≤ Cβ . Let = {ξ ∈ R : |ξ | ≥ R}. Then
∞ 2 2β ix·ξ α −2j − 3ν t ˆ ξ χ3 (ξ )Gλ Rm dξ aj |ξ | t e 3 e Dξ R j =1 ∞ 2 β β − 3ν t |α|−|β1 | −2j 2 ˆ 3 ≤e Rm |ξ | aj |ξ | t Dξ Gλ Dξ dξ β +β +β =2β j =1 1 2 3 ≤ Cα,β e−bt |ξ |−4 dξ ≤ Cβ e−bt . (5.24)
Now consider Aˆ k,l when 2 ≤ k, l ≤ 4 or k, l ∈ {1, 5}. For 2 ≤ k, l ≤ 4, ∞ 2 2β 2,2 eix·ξ Dξ ξ α χ3 (ξ )ξk ξl pj (t)|ξ |−2j e− 3ν t dξ R3 j = |α| 2 +3 −2 |α| 2 +3 −|β1 |−2−|β2 | ≤ Cα e−bt |ξ ||α|+2−|β1 | |ξ | dξ β +β =2β 1 2
≤ Cα,β e−bt
|ξ |−4−2|β| dξ ≤ Cα,β e−bt .
(5.25)
Similarly, for k, l ∈ {1, 5}, ∞ 2 k,l ix·ξ 2β α −2j − 3ν t −bt e Dξ ξ χ3 (ξ ) pj (t)|ξ | e dξ ≤ Cα,β e . R3 j = |α| +2 2
(5.26)
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
All remaining entries of Aˆ are of the form iξk
∞
617
Pj (t)|ξ |−2j , where Pj (t)’s are
j = |α| 2 +2
polynomials of degree no larger than
|α| + 1. If |β| ≥ 1, 2
∞ 2 ix·ξ 2β α −2j − 3ν t dξ e Dξ ξ χ3 (ξ )iξk Pj (t)|ξ | e R3 j = |α| 2 +2 −2 |α| 2 +2 −|β2 | ≤ Cα e−bt |ξ ||α|+1−|β1 | |ξ | dξ β +β =2β 1 2
≤ Cα,β e
−bt
|ξ |−5 dξ ≤ Cα,β e−bt .
(5.27)
When β = 0, we need to estimate ∞ 2 eix·ξ ξ α χ3 (ξ )iξk Pj (t)|ξ |−2j e− 3ν t dξ p.v. R3 j = |α| 2 +2 2 ≤ p.v. eix·ξ ξ α iξk P |α| +2 (t)|ξ |−|α|−4 e− 3ν t dξ 2 ∞ 2 ix·ξ α −2j − 3ν t + e ξ iξk Pj (t)|ξ | e dξ . j = |α| +3 2
The second term on the right-hand side of the above inequality can be estimated in a similar way as before, ∞ 2 Pj (t)|ξ |−2j e− 3ν t dξ eix·ξ ξ α iξk j = |α| 2 +3 |α|+1−2 |α| 2 +3 ≤ Cα e−bt |ξ | dξ ≤ Cα e−bt . (5.28)
The first term requires a more careful estimate. For N > R, we let N = {ξ ∈ R3 : |ξ | ≥ N}. Then 2 p.v. eix·ξ ξ α iξk P |α| +2 (t)|ξ |−|α|−4 e− 3ν t dξ 2 2 = lim eix·ξ ξ α iξk P |α| +2 (t)|ξ |−|α|−4 e− 3ν t dξ. 2 N→∞ \N Note that eix·ξ ξ α iξk |ξ |−|α|−4 dξ = (cos(x · ξ ) + i sin(x · ξ ))ξ α iξk |ξ |−|α|−4 dξ. \N \N
618
D.L. Li
Since |α| is even, we know cos(x · ξ )ξ α ξk |ξ |−|α|−4 is an odd function of ξ . Also, \N is a bounded symmetric region. Therefore, cos(x · ξ )ξ α iξk |ξ |−|α|−4 dξ = 0. \N Hence
\N
eix·ξ ξ α iξk |ξ |−|α|−4 dξ = −
=−
π
π
0
=−
0
2π
N
2π
R N
0
0
R
\N
sin(x · ξ )ξ α ξk |ξ |−|α|−4 dξ
sin (|x|r cos ψ(φ, θ )) r |α|+1−|α|−4 r 2 Q(φ, θ )drdθ dφ sin (|x|r cos ψ(φ, θ )) Q(φ, θ )drdθ dφ, r
where Q(φ, θ ) is a monomial in sin θ , cos θ , sin φ, cos φ. Thus |Q(φ, θ )| ≤ 1. Observe that N N|x|r cos ψ(φ,θ) sin cos ψ(φ, θ )) sin u (|x|r dr = du ≤ C, r u R R|x|r cos ψ(φ,θ) ∞
where C is independent of φ, θ , |x|, N and R, because 0 sin u/udu = π/2. Thus, ix·ξ α −|α|−4 e ξ iξk |ξ | dξ \N π 2π N sin (|x|r cos ψ(φ, θ )) dr ≤ |Q(φ, θ )| dθ dφ ≤ C. r 0 0 R The above inequality implies that 2 ix·ξ α −|α|−4 − 3ν t p.v. ≤ Cα e−bt . e ξ iξ P (t)|ξ | e dξ |α| k +2
2
Therefore, ∞ 2 ix·ξ α −2j − 3ν t e ξ χ3 (ξ )iξk Pj (t)|ξ | e dξ ≤ Cα e−bt . p.v. 3 R j = |α| +2
(5.29)
2
We have now established (5.23) by (5.24), (5.25), (5.26), (5.27) and (5.29).
By (5.20), (5.21) and Lemma 5.16, we have the following proposition. |α|+3 Proposition 5.17. Dxα (χ3 (D)(Gr (x, t) − Fα (x, t))) ≤ Cα,N t − 2 BN (|x|, t). Finally, we shall prove the main result of this paper, Theorem 1.1. Proof of Theorem 1.1. We can write G(x, t) − χ3 (D)Fα (x, t) = GD + Ge + χ1 (D)Gr (x, t) + χ2 (D)Gr (x, t) + χ3 (D)(Gr (x, t) − Fα (x, t)). The theorem is therefore proved by (4.2), Lemma 4.2, (5.5), Propositions 5.10, 5.14 and 5.17.
Green’s Function of Navier-Stokes Equations for Gas Dynamics in R3
619
Acknowledgements. I would like to thank Professor Tai-Ping Liu. Without his patient explanations and thoughtful suggestions, none of this would have happened.
References 1. Alfors, L.V.: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. New York: McGraw-Hill, second edition, 1966 2. Evans, L.C.: Paritial Differential Equations. Volume 19 of Graduate Studies in Mathematics. Providence, Rhode Island: American Mathematical Society, 1998 3. Hoff, D., Zumbrun, K.: Pointwise decay estimates for multidimensional Navier-Stokes diffusion Waves. Z. angew. Math. Phys. 48, 597–614 (1997) 4. Kawashima, S.: Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics. PhD thesis, Kyoto University, 1983 5. Liu, T.-P., Wang, W.: The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Commun. Math. Phys. 196, 145–173 (1998) 6. Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Memoirs of the American Mathematical Society, 125, January 1997, pp. 599 7. Zeng, Y.: L1 Asymptotic Behavior of Compressible Isentropic Viscous 1-D Flow. Commun. Pure Appl. Math. 47, 1053–1082 (1994) Communicated by P. Constantin
Commun. Math. Phys. 257, 621–640 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1352-3
Communications in
Mathematical Physics
Stability of Equilibria with a Condensate Marco Merkli1,2, 1
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montreal, QC, Canada, H3A 2K6. E-mail: [email protected] 2 Centre de Recherches Math´ematiques, Universit´e de Montr´eal, Succursale centre-ville, Montr´eal, QC, Canada, H3C 3J7 Received: 17 May 2004 / Accepted: 13 January 2005 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: A quantum system composed of a spatially infinitely extended free Bose gas with a condensate, interacting with a quantum dot, which can trap finitely many Bosons, has multiple equilibria at fixed temperature. We extend the notion of “return to equilibrium” to systems possessing a multitude of equilibrium states and show that the above system returns to equilibrium in a weak coupling sense: any local perturbation of an equilibrium state converges in the long time limit to an asymptotic state. The latter is, modulo an error term, an equilibrium state which depends, in an explicit way, on the initial local perturbation. The error term vanishes in the small coupling limit. We deduce this stability result from properties of structure and regularity of eigenvectors of the Liouville operator, the generator of the dynamics. Among our technical results is a virial theorem for Liouville type operators which has new applications to systems with and without a condensate. 1. Introduction We study the long-time behaviour of initial states close to an equilibrium state of a Bose gas coupled to a small system that can store a finite number of Bosons (a quantum dot). The Bose gas is so dense (for fixed temperature) or so cold (for fixed density) that it has a Bose-Einstein condensate, inducing long-range correlations. The system possesses many equilibrium states (at fixed temperature T = 1/β). The set of equilibrium states is the convex hull of extremal points, and it is reasonable to expect (and proven for the model considered here) that each extremal equilibrium state has the property of return to equilibrium. This leads to a general definition of this property which we introduce in Sect. 1.1. A feature of this situation is that, starting with a local perturbation of a given superposition of extremal equilibrium states, the system converges in the long time limit to a different superposition of extremal equilibrium states. The asymptotic state depends
Supported by a CRM-ISM postdoctoral fellowship and by McGill University
622
M. Merkli
on the initial condition. In the concrete model discussed here we view this as a limitation of the dispersiveness of the infinite system, caused by long-range correlations. We prove in this paper weak coupling return to equilibrium for the Bose gas with a condensate interacting with a quantum dot: any initial condition close to an equilibrium state of the coupled system, evolving under the coupled dynamics, converges in the long time limit to an asymptotic state. Up to an error term, the latter is again an equilibrium state (different from the initial one), and the error term disappears in the small coupling limit. We expect a stronger result to hold, namely that any locally perturbed equilibrium state approaches some equilibrium state in the large time limit, for small but fixed coupling. This result, called return to equilibrium, has been obtained for systems without a condensate in a variety of recent papers, [JP1, BFS, M, DJ, FM2]. (For a scattering approach to similar problems we refer to [R, LV], and to [HL, QV] for stochastic methods.) It is surprising that none of the methods developed in these references – nor elsewhere, according to our knowledge – seem to be applicable to the present case (which is not a pathological, but a physically relevant one! See also [DWRN]). This is due to the fact that the form factor of the interaction, a coupling function g ∈ L2 (R3 , d 3 k), has the infrared behaviour 0 < |g(0)| < ∞. It lies in between the two “extreme” behaviours g(0) = 0 and |g(0)| = ∞, which are the ones treatable so far. We give here a partial remedy to this situation by establishing a “positive commutator theory” (a first step in a Mourre theory) which is applicable to a wide variety of interactions, including the case where g(0) is a nonzero, finite constant. Our remedy is only partial, we can show return to equilibrium only in the weak coupling sense mentioned above. The obstruction to obtain the stronger result seems to be of technical nature.
1.1. An extended notion of return to equilibrium. Our guiding example is a reservoir of non-interacting Bosons, where the non-uniqueness of KMS states in the condensate phase is due to spontaneous gauge-group symmetry breaking. The kinematical algebra describing the Bose gas is the Weyl algebra W(D) over a test-function space of oneparticle wave functions D ⊂ L2 (R3 , d 3 k). The algebra is generated by Weyl operators W (f ), f ∈ D, satisfying the canonical commutation relations (CCR) W (f )W (g) = i e− 2 Imf,g W (f + g), where ·, · is the inner product of L2 (R3 , d 3 k). The dynamics of the Bose gas is given by the Bogoliubov transformation W (f ) → αt (W (f )) = W (eitω f ), with ω(k) = |k|2 or |k|.
(1)
The first choice in (1) describes non-relativistic, the second one massless relativistic Bosons. Our method can be modified to accommodate for other dispersion relations. As is well known [A] a state ω on W(D) is entirely determined by its generating functional D f → E(f ) := ω(W (f )). The generating functional of the infinitely extended bose gas in equilibrium at inverse temperature β, with an average particle d3k density ρ ≥ ρcrit (β) = (2π)−3 eβω (critical density) is given by −1 1 1 2 3 Eβ,ρ (f ) = exp − f exp − f, (2π) ρf exp −4π 3 ρ0 |f (0)|2 , 4 2
(2)
1 where ρ0 = ρ − ρcrit (β) ≥ 0 is the condensate density, and ρ(k) = (2π )−3 eβω(k) −1 (momentum density distribution of black body radiation according to Planck’s law). D consists of square integrable functions for which (2) exists. Expression (2) has been
Stability of Equilibria with a Condensate
623
obtained in [LP] (see also [AW]) as the thermodynamic limit of grand-canonical finitevolume expectation functionals with fixed mean density ρ. (The canonical case has been considered in [C] and [AW].) Fix ρ0 > 0. The state ωβ corresponding to (2) is not a factor state, that is, the von Neumann algebra of observables represented in the Hilbert space associated to (W(D), ωβ ) is not a factor. This can easily be seen by noticing that ω is not strongly mixing with respect to space translations, see e.g. [BRII, Sect. 5.2.5] and [Ha, Theorem 3.2.2]. To find the factorial decomposition of the state ωβ we analyze the GNS representation of (W(D), ωβ ), [LP, H], which we call here (H2 , π2 , 2 ) (the index 2 will be complemented below by an index 1 referring to the quantum dot). The GNS Hilbert space is ⊕ 2 2 dµβ,ρ (ξ ) F ⊗ F, (3) H2 = F ⊗ F ⊗ L (R , dµβ,ρ ) = R2
where F = F(L2 (R3 , d 3 k)) is the Bosonic Fock space over L2 (R3 , d 3 k), and dµβ,ρ (ξ ) is the probability measure (2πρ0 )−1 e−(r−ρcrit )/ρ0 drdθ with support {ξ = (r, θ ) ∈ [ρcrit ∞) × S 1 } ⊂ R2 . The GNS vector is ⊕ dµβ,ρ (ξ ) F ⊗ F , (4) 2 = F ⊗ F ⊗ 1 = R2
where F is the vacuum in F and 1 is the constant function in L2 (R2 , dµβ,ρ ). The representation map π2 : W(D) → B(H2 ) is given by ⊕ √ ξ dµβ,ρ (ξ )π2 (W (f )), π2 (W (f )) = WF ( 1 + ρf ) ⊗ WF ( ρf ) ⊗ e−i (f,ξ ) = R2
(5)
where WF (f ) = are the Weyl operators in Fock representation (see also (48) below) and the phase ∈ R is given by
(f, ξ ) = (2π)−3/2 2(r − ρcrit ) (Ref (0)) cos θ + (Imf (0)) sin θ , (6) eiϕF (f )
ξ
for ξ = (r, θ ) ∈ [ρcrit , ∞) × S 1 . The fiber of the representation map π2 is π2 (W (f )) = e−i (f,ξ ) π0 (W (f )), where π0 : W(D) → B(F ⊗ F) is the representation √ π0 (W (f )) = WF ( 1 + ρf ) ⊗ WF ( ρf ), (7) referred to as the Araki-Woods representation in [JP1, BFS, DJ, M, FM2]. Relations ξ (3)–(7) give the decomposition ωβ = R2 dµβ,ρ (ξ )ωβ , where ωβ (W (f )) = e−i (f,ξ ) F ⊗ F , π0 (W (f ))F ⊗ F ξ
(8)
is easily seen to be a factorial (hence extremal) β-KMS state with respect to the dynamics (1). The gauge transformations γs (W (f )) = W (eis f ), s ∈ R, commute with the ξ dynamics, αt ◦γs = γs ◦αt for all s, t ∈ R, and while ωβ is invariant under γs , the ωβ are not: the gauge-group symmetry is spontaneously broken, leading to an S 1 -multitude of KMS states, represented by the angular variable of ξ = (r, θ ) (the variable r comes from ξ the use of the grand-canonical ensemble). The states ωβ satisfy the property of return to equilibrium: limt→∞ ωβ (B ∗ αt (A)B) = ωβ (B ∗ B)ωβ (A), for any A, B ∈ W(D). This motivates the following ξ
ξ
ξ
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Definitions. Let ωξ , ξ ∈ X (a measurable space), be families of states on a C ∗ algebra A, and denote their GNS representations by (Hξ , πξ , ξ ). Suppose that, for each ξ , σξt is a ∗automorphism group of the von Neumann algebra πξ (A) , such that ξ → σξt (πξ (A))πξ (B)ξ is measurable, for all A, B ∈ A, t ∈ R (and where ·ψ = ψ, ·ψ). 1. We say that the family ωξ is asymptotically stable (w.r.t. σξt ) if, for any probability measure µ on X, and any A, B ∈ A, we have t lim dµ(ξ ) σξ (πξ (A)) = dµ(ξ ) ωξ (B ∗ B) ωξ (A). (9) t→∞ X
πξ (B)ξ
X
2. If the ωξ are (β, σξt )-KMS states of πξ (A) we say the family ωξ has the property of return to equilibrium. (Typically, one will choose ωξ to be the family of extremal KMS states for a given system.) The definitions can be extended to the case where µ is a probability measure on the space of all states on A. If ωξ is an asymptotically stable family, and B satisfies ωξ (B ∗ B) = 1 for all ξ (say B is unitary) then the asymptotic state is independent of B. In general, the effect of the initial condition on the limit state is a redistribution of the relative weights. In case σξt (πξ (A)) = πξ (αt (A)) for some ∗automorphism group αt of A (as for the free Bose gas), one can formulate the above definitions on a purely C ∗ algebraic level1 . In this paper we show a version of (9), where the ωξ are extremal equilibrium states of the Bose gas with a condensate interacting with a quantum dot, and where the time limit is taken in the ergodic mean sense and is followed by the limit of small coupling, see (26). 2. Model and Main Results Quantum tweezers consist of a supercritical Bose gas interacting with a small system which can trap finitely many Bosons. One can imagine the use of such a trap to remove single (uncharged) particles from a reservoir (condensate), hence the name quantum tweezers, [DWRN]. The C ∗ algebra for the quantum tweezers is A = B(Cd ) ⊗ W(D),
(10)
the non-interacting dynamics is the ∗automorphism group α0t = α1t ⊗ α2t
(11)
of A, where we now denote the free field dynamics (1) by α2t , and where A → α1t (A) = eitH1 Ae−itH1 is the quantum dot dynamics, generated by the Hamiltonian H1 = diag(0, 1, 2, . . . , d − 1).
(12)
(Our method applies to any selfadjoint diagonal matrix with non-degenerate spectrum.) The vector [1, 0, . . . , 0] ∈ Cd represents the ground state (no particle trapped), 1 The measurability condition in the definition above can then be reduced to one not involving the dynamics αt , as has also been pointed out by the referee.
Stability of Equilibria with a Condensate
625
[0, 1, 0, . . . , 0] the first excited state (one particle trapped) of the quantum dot, etc. The raising operator, G+ , has matrix elements (G+ )j k = δj,k+1 , j, k = 1, . . . , d
(13) ∗,
(G+ has ones on its subdiagonal), and the lowering operator is G− = G+ so that H1 G± = G± (H1 ± 1). The action of G+ (G− ) increases (decreases) the excitation level by one. Let µ be a fixed probability measure on [ρcrit , ∞)×S 1 and consider the (β, α0t )-KMS state ξ ξ ξ ωβ,0 = dµ(ξ ) ωβ,0 , ωβ,0 = ω1,β ⊗ ωβ . (14) R2
Here, ω1,β is the (β, α1t )-KMS state (Gibbs state) of the small system, determined by the ξ density matrix ρβ = e−βH1 /tr e−βH1 , and ωβ is a (β, α2t )-KMS state with fixed density and phase, (8). The subindex “0” in (14) indicates the absence of an interaction. The GNS Hilbert space H associated to the algebra A, (10), and the state (14) is ⊕ H= dµ(ξ ) Hξ , Hξ = Cd ⊗ Cd ⊗ F ⊗ F, (15) R2
ξ
where the fiber Hξ is the representation Hilbert space of (A, ωβ,0 ). The cyclic vector β,0 representing the state (14) is ⊕ ξ ξ β,0 = dµ(ξ ) β,0 , β,0 = 1,β ⊗ F ⊗ F , (16) R2
and 1,β is the Gibbs vector (H1 ϕj = Ej ϕj ) 1,β = √
1 tr e−βH1
e−βEj /2 ϕj ⊗ ϕj .
(17)
j
The representation map is given by ⊕ ξ π= dµ(ξ ) πξ , πξ = π1 ⊗ π2 , R2
(18)
ξ
where π1 (A) = A ⊗ 1lCd , A ∈ B(Cd ), and π2 is defined after (6). We introduce the von Neumann algebra ⊕ dµ(ξ ) M1 ⊗ M0 ⊂ B(H), (19) Mβ = π(A) = R2
where M1 = B(Cd ) ⊗ 1lCd and M0 = π0 (W(D)) ⊂ B(F ⊗ F). Remark. It is well known that the von Neumann algebra M0 is a factor. One can show ⊕ ξ that the representation π2 := R2 dµ(ξ )π2 has the property M2 := π2 (W(D)) = M0 ⊗ M, where M is the abelian von Neumann algebra of all multiplication operators on L2 (R2 , dµ) (see also (5)), satisfying M = M. The centre of M2 is Z(M2 ) = 1lF ⊗F ⊗ M, so M2 and hence Mβ are not factors. Equation (19) is the central decomposition of Mβ .
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M. Merkli
The direct integral decomposition reduces the description of the system with a conξ densate to one of a family of systems of no-condensate type, labelled by ξ . Since ωβ,0 is invariant under α0t there is a selfadjoint operator L0,ξ acting on Hξ , called the (standard) Liouvillian, satisfying πξ (α0t (A)) = eitL0,ξ πξ (A)e−itL0,ξ , ξ L0,ξ β,0
(20)
= 0,
(21)
for all A ∈ A. The r.h.s. of (20) extends to a ∗automorphism group of Mβ which we t . We shall not burden the reader with further explicit expressions at this point, call σ0,ξ they will be given in Sect. 3.1. The interacting dynamics σλt on Mβ is a perturbation of ⊕ t , obtained by replacing the generator L = ⊕ dµ(ξ )L σ0t = R2 dµ(ξ )σ0,ξ 0 0,ξ by R2 Lλ = L0 + λI,
(22)
where λ ∈ R is a coupling constant and I is the operator on H determined by the formal interaction term
λ G+ ⊗ a(g) + G− ⊗ a ∗ (g) . (23) G± are the raising and lowering operators, (13), and a # (g) are creation (# = ∗) and annihilation operators of the heat bath, smeared out with a function g ∈ D, called a form factor. The operator G+ ⊗ a(g) destroys a Boson and traps it in the quantum dot (whose excitation level is increased by one) and similarly, the effect of G− ⊗a ∗ (g) is to release a Boson from the quantum dot. The total number of particles is preserved ((23) commutes with H1 + R3 a ∗ (k)a(k)d 3 k). Since the quantum dot can absorb only finitely many Bosons, the interacting equilibrium state is a (local) perturbation of the non-interacting one. A physically different situation occurs when the condensate is coupled to another reservoir. We would then expect that time-asymptotic states are of non-equilibrium stationary nature. Of course, (23) has a meaning only in a regular representation of the Weyl algebra, e.g. the representation π above, see Sect. 3.1. The generator Lλ is reduced by the direct integral decomposition, and by the structural stability of KMS states (applied here to each fixed fiber ξ , see Sect. 3.1), we have ⊕ ⊕ ⊕ ξ t β,λ = dµ(ξ )β,λ , Lλ = dµ(ξ )Lλ,ξ , σλt = dµ(ξ )σλ,ξ , (24) R2
R2
R2
ξ
t , the ∗automorphism group on M ⊗ M generated where β,λ is a KMS vector for σλ,ξ 1 0 ξ
by Lλ,ξ , with the property Lλ,ξ β,λ = 0. We make two assumptions on the form factor g determining the interaction. (A1) Regularity. The form factor g is a function in C 4 (R3 ) and satisfies √ √
(1 + 1/ ω)(k · ∇k )j ( ρ + 1 + ρ)g L2 (R3 ,d 3 k) < ∞, for j = 0, . . . , 4, and (1 + ω)2 g L2 (R3 ,d 3 k) < ∞. (A2) Effective coupling. We assume that S 2 dσ |g(1, σ )|2 = 0. Here, g is represented in spherical coordinates.
Stability of Equilibria with a Condensate
627
Theorem 1 (Weak coupling return to equilibrium). Assume conditions (A1) and (A2). Let > 0, ξ ∈ [ρcrit , ∞) × S 1 , B ∈ A be fixed. There is a λ0 (, ξ, B) > 0 s.t. if 0 < |λ| < λ0 (, ξ, B) then T ξ ξ ξ ∗ lim 1 < A , (25) dt σ (π (A)) − ω (B B)ω (A) ξ t,λ ξ β,λ β,λ T →∞ T πξ (B)β,λ 0 ξ for all A ∈ A, where the coupled KMS state is ωβ,λ (·) = πξ (·) ξ , (18), (24). β,λ
µ
Let µ be any probability measure supported on [ρcrit , ∞)×S 1 and consider ωB (σλt (A)) ξ := R2 dµ(ξ )σt,λ (πξ (A))π (B)ξ (where σλt (A) has to be understood cum grano salis ξ
β,λ
in this expression). It follows from (25) that 1 T µ ξ ξ lim lim dt ωB (σλt (A)) = dµ(ξ ) ωβ,0 (B ∗ B)ωβ,0 (A), 2 λ→0 T →∞ T 0 R ξ ωβ,0 (·) = πξ (·) ξ being the uncoupled KMS state, (16), (18).
(26)
β,0
Remarks. 1. We expect the r.h.s. of (25) to be zero for λ sufficiently small. The obstruction to showing this by existing strategies is that they all need the condition that either g(0) = 0, or |g(k)| → ∞, as |k| → 0. The first case is uninteresting in the presence of a condensate (no coupling to the modes of the condensate, see the next remark), and the second type of form factor does not enter into the description of a system with a condensate (the form factor would not be in the test function space D). The technical reason why our method does not yield the stronger result is that due to the infrared behaviour g(0) = 0 we must use a conjugate operator A giving rise to a (positive) commutator which does not have a spectral gap at zero (cf. (28), (27)). 2. Condition (A2) is often called the Fermi Golden Rule Condition. A heuristic calculation shows that the probability for the process of trapping a Boson in the state f in the λ2 quantum dot, to second order in λ, and for large times, is ∝ (eβω(1) |f (1)g(1)|2 (f, g −1)2 radial functions, Bose gas at critical density, ρ0 = 0), and ∝ (1 − cos t)λ2 ρ02 |f (0)g(0)|2 (density ρ0 fixed and β very large, i.e. almost pure condensate).
To state the virial theorem and to measure regularity of eigenvectors of Lλ,ξ , (24) (see also (51) below), we introduce the non-negative selfadjoint operator = d(ω) ⊗ 1lF + 1lF ⊗ d(ω),
(27)
where d(ω) is the second quantization of the operator of multiplication by ω(k) on L2 (R3 , d 3 k), cf. (1). The kernel of is spanned by the vector 0 = F ⊗ F and has no nonzero eigenvalues. The operator represents the quadratic form i[L0 , A], the commutator of L0 with the conjugate operator
3
A = d(ad ) ⊗ 1lF − 1lF ⊗ d(ad ),
(28)
ad = i k · ∇k + 2 being the selfadjoint generator of dilations on L2 (R3 , d 3 k). The formal relation = i[L0 , A] follows from i[ω, ad ] = ω (for relativistic Bosons, see (1); in the non-relativistic case we have i[ω, ad /2] = ω). Let Cj be the selfadjoint operators representing the j -fold commutator of Lλ,ξ with iA (defined as quadratic forms, cf. Sect. 3.1).
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M. Merkli
Theorem 2 (Virial Theorem, regularity of eigenvectors of Lλ,ξ ). Assume Condition (A1) and let ξ ∈ [ρcrit , ∞) × S 1 be fixed. If ψ is an eigenfunction of Lλ,ξ then ψ is in the form domain of C1 , and ψ, C1 ψ = 0.
(29)
There is a constant c which does not depend on ψ, ξ nor on β ≥ β0 , for any β0 > 0 fixed, such that
1/2 ψ ≤ c|λ| ψ .
(30)
Remark. The regularity bound (30) follows easily from (29) and the fact that C1 = + λI1 , where I1 is infinitesimally small relative to , cf. (60), so that 0 = ψ, C1 ψ ≥ 2 (1 − ) ψ, ψ − λ c ψ 2 , for any > 0, for some constant c independent of ξ and β, as mentioned in the theorem. We refer for a more complete exposition of this to [FM1]. We prove Theorem 2 in Sect. 3.1 by showing that the hypotheses leading to our Theorem 8 below, a more general result, are satisfied in the present situation. Our next result describes the structure of kerLλ,ξ . Let P ( ≤ x) stand for the spectral projection of onto the interval [0, x]. Theorem 3 (Structure of the kernel of Lλ,ξ ). Assume Conditions (A1), (A2) and let ξ ∈ [ρcrit , ∞) × S 1 be fixed. There is a number λ0 > 0 s.t. if 0 < |λ| < λ0 then any normalized ψ ∈ ker(Lλ,ξ ) satisfies
P1,β P ( ≤ |λ|)ψ ≥ 1 − O(λ0 ),
(31)
where P1,β is the projection onto C1,β (see (17)) and O(λ0 ) is a vector whose norm, which is independent of ψ, tends to zero in the limit λ → 0 (uniformly in ξ in any compact set and in β ≥ β0 , for any β0 > 0 fixed). The constant λ0 is uniform in ξ in any compact set, and in β ≥ β0 , for any fixed β0 > 0. Our proof of this theorem, given in Sect. 5, relies on a positive commutator estimate and Theorem 2. Expansion (31) implies that the only vector in the kernel of Lλ,ξ which ξ does not converge weakly to zero, as λ → 0, is the interacting KMS state β,λ , (24). This information on the kernel of Lλ,ξ alone enters our proof of Theorem 1. ξ
Corollary 4. Assume (A1) and (A2) and let Pβ,λ be the projection onto the subspace ξ β,λ ,
(24). Let ξ ∈ [ρcrit , ∞) × S 1 be fixed. Any spanned by the interacting KMS state
ξ ⊥ normalized element ψ ∈ ker(Lλ,ξ ) ∩ Ran Pβ,λ converges weakly to zero, as λ → 0. The convergence is uniform in ξ in any compact set and in β ≥ β0 , for any β0 > 0 fixed. We prove the corollary in Sect. 5. The virial theorem we present in Sect. 3, Theorem 8, is applicable to systems without a condensate, in which case one is interested in form factors g which have a singularity at the origin. Theorem 8 is therefore relevant in the study of return to equilibrium and thermal ionization for systems without condensate, as will be explained in [FM3].
Stability of Equilibria with a Condensate
629
Theorem 5 (Improved Virial Theorem for systems without condensate). Let Lλ be the Liouvillian of a system without condensate, Lλ = L0 + λI (i.e., Kξ = 0 in (52)) and suppose that the form factor g is in C 4 (R3 \{0}) and satisfies the condition √ √ √ (32) (1 + 1/ ω)(ad )j 1 + ρ g, (1 + 1/ ω)(ad )j ρ g ∈ L2 (R3 , d 3 k), 2 j 2 j√ 2 3 3 (1 + ω) (ad ) 1 + ρ g, (1 + ω) (ad ) ρ g ∈ L (R , d k), (33) for j = 0, . . . , 4. Then the conclusions (29), (30) of Theorem 2 hold. An admissible infrared behaviour of g satisfying (32), (33) is g(k) ∼ |k|p , as |k| ∼ 0, with p > −1/2 for relativistic Bosons (cf. (1)). The range of treatable values of p obtained in previous works, [M, DJ, FM1, FM2], is p = −1/2, 1/2, 3/2, p > 2. Theorem 5 fills in the gaps between the discrete values of admissible p. Theorem 6. Assume the setting of Theorem 5, that (A2) holds and that |g(k)| ≤ c|k|p , for |k| < c , for some constants c, c , and where p > −1/2 (for relativistic Bosons, and p > 21 for nonrelativistic ones). The conclusion (31) of Theorem 3 holds. 2.1. Proof of Theorem 1, given Corollary 4. Fix η > 0, ξ ∈ [ρcrit , ∞) × S 1 , B ∈ A and choose an element bξ,η ∈ πξ (A) s.t. πξ (B)β,0 − bξ,η β,0 = O(η) (β,0 is cyclic 0 for the commutant πξ (A) ). It follows that πξ (B)β,λ − bξ,η β,0 = O(η + B λ ) ξ
ξ
ξ
(because β,λ − β,0 = O(λ0 ), see [FM2]), and consequently t σλ,ξ (πξ (A)) = πξ (B ∗ )bξ,η eitLλ,ξ πξ (A) ξ πξ (B)β,λ
ξ
β,λ
+ R1 ,
(34)
)λ0 , where we use that b with R1 = O A B η + ( B + bξ,η ξ,η commutes ξ
t (π (A)), and that L with σλ,ξ ξ λ,ξ β,λ = 0. The von Neumann ergodic theorem tells us 1 T that the ergodic average, T 0 dt, of the first term on the r.h.s. of (34) converges in the limit T → ∞ to ξ λ,ξ πξ (A) ξ = πξ (B ∗ )bξ,η ωβ,λ (A) πξ (B ∗ )bξ,η ξ β,λ
+
β,λ
∞
ξ ξ ξ ξ ψj,λ ψj,λ , πξ (A)β,λ , πξ (B)β,λ , bξ,η
(35)
j =1 ξ
ξ
where λ,ξ is the projection onto the kernel of Lλ,ξ for which β,λ ∪ {ψj,λ }j ≥1 is an orthonormal basis. It follows from Corollary 4 that the series in (35) converges to zero as λ → 0. Moreover we have
ξ ∗ 0 , (36) = ω (B B) + O
B
η + ( B
+
b
)λ πξ (B ∗ )bξ,η ξ,η ξ β,λ β,λ
where we use the estimates given at the beginning of the proof. From (34), (35), (36) it follows that there exists a λ1 (ξ, η) > 0 s.t. if 0 < |λ| < λ1 (ξ, η) then 1 T t ξ ξ ∗ σλ,ξ (πξ (A)) ωβ,λ (A) + R3 , (37) lim = ω (B B) + R 2 ξ β,λ T →∞ T 0 πξ (B)β,λ
630
M. Merkli
)λ0 where R2 = O B η + ( B + bξ,η and R3 = O A B η + ( B + )λ0 . Given > 0 (as in the theorem) we can choose first η small and then λ
bξ,η small, in such a way that R2 < /2 and R3 < A /2. 3. Another Abstract Virial Theorem with Concrete Applications We give a virial theorem in an abstract setting covering the cases of interest in the present paper. The virial theorem developed in [FM1], where the dominant part of [L, A] commutes with A, does not apply to the present situation (L = Lλ,ξ ); here the leading term of [[L, A], A] is L. Let H be a Hilbert space, D ⊂ H a core for a selfadjoint operator Y ≥ 1l, and X a symmetric operator on D. As in [FM1] we say the triple (X, Y, D) satisfies the GJN (Glimm-Jaffe-Nelson) Condition, or that (X, Y, D) is a GJN-triple, if there is a constant k < ∞, s.t. for all ψ ∈ D:
Xψ ≤ k Y ψ and ± i {Xψ, Y ψ − Y ψ, Xψ} ≤ k ψ, Y ψ .
(38)
Theorem 7 (GJN commutator theorem). If (X, Y, D) satisfies the GJN Condition, then X determines a selfadjoint operator (again denoted by X), s.t. D(X) ⊃ D(Y ). Moreover, X is essentially selfadjoint on any core for Y , and the first bound in (38) is valid for all ψ ∈ D(Y ). Suppose one is given a selfadjoint operator Y ≥ 1l with core D ⊂ H, and operators L, A, ≥ 0, D, Cn , n = 0, . . . , 4, all symmetric on D, satisfying ϕ, Dψ = i {Lϕ, ψ − ϕ, Lψ} , (39) C0 = L, ϕ, Cn ψ = i {Cn−1 ϕ, Aψ − Aϕ, Cn−1 ψ} , n = 1, . . . , 4, (40) where ϕ, ψ ∈ D. Assume that (VT1) (X, Y, D) satisfies the GJN Condition, for X = L, , D, Cn . (Hence we view these as selfadjoint operators). (VT2) A is selfadjoint, D ⊂ D(A), eitA leaves D(Y ) invariant. (VT3) D ≤ k1/2 in the sense of Kato on D, for some constant k. (VT4) Let the operators Vn be defined as follows: for n = 1, 3 set Cn = + Vn , and set C2 = L2 + V2 , C4 = L4 + V4 . The following relative bounds hold in the sense of Kato on D: Vn ≤ k1/2 , for n = 1, . . . , 4, L4 ≤ k, L2 ≤ kr , for some r > 0.
(41) (42) (43)
We prove the following virial theorem in Sect. 4. Theorem 8 (Virial Theorem). We assume the setting and assumptions introduced in this section so far. If ψ ∈ H is an eigenvector of L then ψ is in the form domain of C1 and C1 ψ = 0.
(44)
Stability of Equilibria with a Condensate
631
3.1. Concrete applications: Proofs of Theorems 2 and 5. Let us start by obtaining explicit expressions for the Liouvillian. It is well known and easy to verify that the operator L0,ξ = L0 introduced in (20), (21) is given by L0 = L 1 + L 2 , L1 = H1 ⊗ 1lCd − 1lCd ⊗ H1 , L2 = d(ω) ⊗ 1lF − 1lF ⊗ d(ω).
(45) (46) (47)
Here d(ω) is the second quantization of the operator of multiplication by ω on L2 (R3 , d 3 k). We will omit trivial factors 1l or indices Cd , F whenever we have the reasonable hope that no confusion can arise (e.g. L1 really means L1 ⊗1lF ⊗1lF ). The field operator ϕξ (f ) = 1i ∂t |t=0 πξ (W (tf )) in the representation πξ , (18), is easily calculated to be √ ϕξ (f ) = ϕF ( 1 + ρf ) ⊗ 1l + 1l ⊗ ϕF ( ρf ) − (f, ξ ),
(48)
where (f, ξ ) is given in (6), ϕF (f ) = √1 (aF ∗ (f ) + aF (f )), and aF ∗ (f ), aF (f ) are 2 the smeared out creation, annihilation operators satisfying the commutation relations [aF (f ), aF ∗ (g)] = f, g , [aF (f ), aF (g)] = [aF ∗ (f ), aF ∗ (g)] = 0. Our convention is that f → aF (f ) is an antilinear map. We define
∗ √ Vξ = G+ ⊗ 1lCd ⊗ aF 1 + ρg ⊗ 1lF + 1lF ⊗ aF ρg −(2π)−3/2 2(r − ρcrit ) g(0) eiθ + adjoint,
(49)
which corresponds formally to πξ G+ ⊗ a(g) + adjoint (apply (18) to (23)). Vξ is an unbounded selfadjoint operator on Hξ which is affiliated with M1 ⊗ M0 . For t ∈ R, A ∈ M1 ⊗ M0 we set t σλ,ξ (A)
=
n≥0
(iλ)
t
n
dt1 . . . 0
dtn eitn L0 Vξ e−itn L0 , · · ·
tn−1
0
· · · eit1 L0 Vξ e−it1 L0 , A · · · .
(50)
It is a standard thing to show that the series converges in the strong sense on a dense set of vectors, for any A ∈ M1 ⊗ M0 , λ, t ∈ R (see e.g. [FM1]). Since Vξ is affiliated with M1 ⊗ M0 and eitL0 · e−itL0 leaves M1 ⊗ M0 invariant, the integrand in (50) does not change if we add to each eitj L0 Vξ e−itj L0 a term −J eitj L0 Vξ e−itj L0 J = −eitj L0 J Vξ J e−itj L0 , where J is the modular conjugation operator associated to (M1 ⊗ ξ M0 , β,0 ), [DJP, FM1]. In other words, we replace Vξ in (50) by Vξ − J Vξ J . The r.h.s. of (50) is then identified as the Dyson series expansion of the ∗automorphism group t (A) = eitLλ,ξ Ae−itLλ,ξ of M ⊗ M , where the standard, interacting Liouvillian σλ,ξ 1 0 Lλ,ξ is the selfadjoint operator Lλ,ξ = L0 + λIξ .
(51)
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M. Merkli
L0 is given in (45) and we define Iξ = I + K ξ ,
∗
√
(52)
I = G+ ⊗ 1lCd ⊗ aF ( 1 + ρ g) ⊗ 1lF + 1lF ⊗ aF ( ρ g) + adj. √ −1lCd ⊗ C1 G+ C1 ⊗ aF ∗ ( ρg) ⊗ 1lF + 1lF ⊗ aF ( 1 + ρ g) + adj., (53) Kξ = Kξ1 ⊗ 1lCd ⊗ 1lF ⊗F − 1lCd ⊗ C1 Kξ1 C1 ⊗ 1lF ⊗F ,
√ Kξ1 = −2(2π )−3/2 r − ρcrit G+ g(0)eiθ + G− g(0)e−iθ ,
(54) (55)
where C1 is the antilinear map on Cd ⊗ Cd implementing complex conjugation in the basis ϕi ⊗ ϕj diagonalizing L1 (this comes from the explicit form of J , see e.g. [DJP, FM1]). By perturbation theory of KMS states, the vector β,λ = (Zβ,λ )−1 e−β(L0 +λIξ, )/2 β,0 ∈ Hξ , ξ
ξ
ξ
ξ
(56)
ξ
t )-KMS state. I where Zβ,λ is a normalization factor ( β,λ = 1), is a (β, σλ,ξ ξ, is obtained by dropping the second term (the one coming with a minus sign) both in ξ (53) and in (54). The fact that β,0 , (16), is in the domain of the unbounded operator √ e−β(L0 +λIξ, )/2 , provided g/ ω L2 (R3 ) < ∞ (see also (A1)), can be seen by expanding ξ the exponential in a Dyson series and verifying that the series applied to β,0 converges, see e.g. [BFS]. Furthermore, by the choice Vξ → Vξ − J Vξ J in (50) explained above we achieve that [DJP, FM1] ξ
Lλ,ξ β,λ = 0.
(57)
D = Cd ⊗ Cd ⊗ F0 (C0∞ (R3 , d 3 k)) ⊗ F0 (C0∞ (R3 , d 3 k)),
(58)
We now verify (VT1)–(VT4). Take
where F0 is the finite-particle subspace of Fock space, set Y = d(ω + 1) ⊗ 1lF + 1lF ⊗ d(ω + 1) + 1l,
(59)
and let the operators L, , A of Sect. 3 above be given, respectively, by the operators Lλ,ξ (see (51), or Lλ in the case of Theorem 5), (27), and (28). We calculate C1 = + λI1 , with √ I1 = G+ ⊗ 1lCd ⊗ aF (ad 1 + ρ g) ⊗ 1lF − 1lF ⊗ aF ∗ (ad ρ g) + adj. √ −1lCd ⊗ C1 G+ C1 ⊗ aF ∗ (ad ρg) ⊗ 1lF − 1lF ⊗ aF (ad 1 + ρ g) + adj. (60) Similarly one obtains expressions C2 = L2 +λI2 , C3 = +λI3 , C4 = L2 +λI4 , where L2 is given in (47), and where the Ij are obtained similarly to I1 , (60). The operator D, (39), is just iλ[I, ]. It is a routine job to verify that assuming (A1) (or (32), (33)), Conditions (VT1)–(VT4) hold, with Vn = In and L4 = L2 , r = 1. To check Condition (VT2) one can use the explicit action of eitA , see also [FM1], Sect. 8.
Stability of Equilibria with a Condensate
633
4. Proof of Theorem 8 Before immersing ourselves into the details of the proof we present some facts we shall use repeatedly. – If a unitary group eitX leaves the domain D(Y ) invariant then there exist constants k, k s.t. Y eitX ψ ≤ kek |t| Y ψ , for all ψ ∈ D(Y ), see [ABG], Props. 3.2.2 and 3.2.5. Moreover, if (X, Y, D) is a GJN triple then the unitary group eitX leaves D(Y ) invariant. – Let (X, Y, D) and (Z, Y, D) be GNS triples, and suppose that the quadratic form of the commutator of X with Z, multiplied by i, is represented by a symmetric operator on D, denoted by i[X, Z], and that (i[X, Z], Y, D) is a GNJ triple. Then we have t eitX Ze−itX − Z = dt1 eit1 X i[X, Z]e−it1 X . (61) 0
This equality is understood in the sense of operators on D(Y ). Of course, if the higher commutators of X with Z also form GJN triples with Y, D then one can iterate formula (61). We refer to [FM1] and the references therein for more detail and further results of this sort. Let us introduce the cutoff functions x 2 2 f1 (x) = dy e−y , f (x) = e−y /2 , (62) −∞
g = g12 ,
(63)
where g1 ∈ C0∞ ((−1, 1)) satisfies g1 (0) = 1. The derivative (f1 ) equals f 2 which is strictly positive and the ratio (f )2 /f decays faster than exponentially at infinity. The Gaussian f is the fixed point of the Fourier transform f(s) = (2π)−1/2 dx e−isx f (x), (64) R
2 which is a Gaussian itself. This 1 = f i.e., f(s) = e , and we have (f 1 ) = is f means that f1 decays like a Gaussian for large |s| and has a singularity of type s −1 at the origin. We define cutoff operators, for ν, α > 0, by g1,ν = g1 (ν) = (2π)−1/2 ds g1 (s)eisν , (65) −s 2 /2
R
gν =
2 g1,ν ,
fα = f (αA) = (2π)−1/2
R
(66) ds f(s)eisαA .
(67)
Since f1 has a singularity at the origin, we cut a small interval (−η, η) out of the real axis, where η > 0, and define η f1,α = α −1 (2π)−1/2 ds f1 (s)eisαA , (68) Rη
where we set Rη = R\(−η, η). Standard results about invariance of domains show that η the cutoff operators gν , fα , f1,α are bounded selfadjoint operators leaving the domain
634
M. Merkli η
D(Y ) invariant, and it is not hard to see that f1,α ≤ k/α, uniformly in η (see [FM1]). Suppose that ψ is a normalized eigenvector of L with eigenvalue e, Lψ = eψ, ψ = 1. Let ϕ ∈ H be s.t. ψ = (L + i)−1 ϕ and let {ϕn } ⊂ D be a sequence approximating ϕ, ϕn → ϕ. Then we have ψn = (L + i)−1 ϕn → ψ, as n → ∞, and ψn ∈ D(Y ). The latter statement holds since the resolvent of L leaves D(Y ) invariant (which in turn is true since (L, Y, D) is a GJN triple). It follows that the regularized eigenfunction ψα,ν,n = fα gν ψn
(69)
is in D(Y ), and that ψα,ν,n → ψ, as α, ν → 0 and n → ∞. It is not hard to see that η (L − e)ψn → 0 as n → ∞, a fact we write as (L − e)ψn = O(n0 ). Since f1,α leaves η D(Y ) invariant, and since D(Y ) ⊂ D(L), the commutator −i[f1,α , L] is defined in the usual (strong) way on D(Y ). We consider its expectation value in the state gν ψn ∈ D(Y ), η η −i [f1,α , L] = −i [f1,α , L − e] . (70) gν ψn
gν ψn
The idea is to write (70) on the one hand as C1 ψα,ν,n modulo some small term for appropriate α, ν, n (“positive commutator”), and on the other hand to see that (70) itself is small, using the fact that (L − e)ψ = 0. The latter is easily seen: (L − e)gν ψn = gν (L − e)ψn + g1,ν [L, g1,ν ]ψn + [L, g1,ν ]g1,ν ψn ,
(71)
and due to condition (VT3), s
√ ν isν g1,ν [L, g1,ν ] = ds g1 (s)e ds1 e−is1 ν g1,ν Deisν = O ν , (2π)1/2 R 0 and similarly, [L, g1,ν ]g1,ν = O
√ ν , so that
η −i [f1,α , L]
gν ψn
=O
O(n0 ) + α
√ ν
.
(72)
Next we get a lower bound on (70). A repeated application of formula (61) gives, in the strong sense on D(Y ), α α 2 η C1 − i f1,α C2 − −i[f1,α , L] = f1,α f C3 2! 3! 1,α s2 s4 s s1 iα 3 isαA + ds f1 (s)e ds1 ds2 ds3 ds4 e−is4 αA C4 eis4 αA (2π )1/2 Rη 0 0 0 0 α α2 Rη,2 C2 + C3 , (73) 2! 3! = where we use that (2π)−1/2 R ds (is)n f(s)eisx = f (n) (x), and where we set f1,α = (f ) (αA), etc., and (f1 ) (αA), f1,α 1 +Rη,1 C1 +
Rη,n = −i(2π)−1/2
η −η
ds s n f1 (s)eisαA .
(74)
Stability of Equilibria with a Condensate
635
= f 2 (αA) = f 2 and applying again expansion (61) yields Using that f1,α α
α2 f1,α C1 = fα C1 fα + iαfα fα C2 + fα fα C3 2! s s1 s2 α3 isαA − fα ds f (s)e ds1 ds2 ds3 e−is3 αA C3 eis3 αA . (2π )1/2 R 0 0 0 (75) = 2f f , then Plug this into the r.h.s. of (73) and use f1,α α α
η −i [f1,α , L]
gν ψn
1 1 C3 fα fα C3 − f1,α 2 3! η η α3 +O +√ + , νr ν ν
= C1 ψα,ν,n + α 2 Re
gν ψn
(76)
where we take the real part on the r.h.s. for free since the l.h.s. is real. The error term in (76) is obtained as follows. Certainly, Rη,n = O (η), and (VT4) gives
condition
√ Cn gν = O ν −r + ν −1/2 , which accounts for the term O η/ν r + η/ ν . The term
O α 3 /ν is an upper bound for the expectation of the terms in (73) and (75) involving the multiple integrals, in the state gν ψn . For instance, the contribution coming from (73) is bounded above as follows. Due to condition (VT4) we have e−is4 αA C4 eis4 αA gν ψn ≤ k eis4 αA gν ψn = ek α|s4 | O ν1 , which gives the following upper bound on the rele
vant term: α 3 Rη ds |f1 (s)|s 4 ek |s| ·O ν1 . The integral is finite because f1 has Gaussian decay. Our next task is to estimate the real part term in (76). It is enough to consider α 2 Re fα fα C3 g ψ and α 2 Re (fα )2 C3 , (77) ν
n
gν ψn
= 2(f )2 + 2f f . Let us start with the first term in (77). Using the because f1,α α α α decompostion C3 = + V3 and the relative bound of V3 given in (VT4) we estimate
α 2 Re fα fα C3 g
ν ψn
α2 √ ν ψn ν 2 α α3 2 . = α Re fα fα g ψ + O √ + ν n ν ν
= α 2 Re fα fα g
+O
We bound the first term on the r.h.s. from above as α 2 Re fα fα g ψ ≤ α 2 1/2 fα gν ψn 1/2 ψα,ν,n
ν
n
(78)
(79)
and use that fα fα g ψ ≤ R ds |f (s)| fα eisαA g ψ = O ν1 to see that for ν n ν n any c > 0, α 2 Re fα fα g
α4 + c ψα,ν,n . ≤ ν ψn cν
(80)
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M. Merkli
Choose c = α 1+ξ , for some ξ > 0 to be determined later. Then, inserting again
a term √ V1 into the last expectation value (by adding a correction of size O α 1+ξ / ν ), we get 2 α 1+ξ α α3 α 3−ξ 1+ξ |(78)| ≤ α C1 ψα,ν,n + O √ + + √ + . (81) ν ν ν ν Next we tackle the second term in (77). The Gaussian f is strictly positive, so we can write 2 2 (fα ) α α3 2 2 2 α Re (fα ) C3 = α Re fα +O √ + , (82) gν ψn fα ν ν gν ψn where we have taken way into account condition (VT4) in the
same as above. It follows (f )2 α2 α3 that |(82)| ≤ α 2 1/2 fαα gν ψn 1/2 ψα,ν,n + O √ + ν , and proceeding as ν in (79)–(80) we see that 2 α3 α 3−ξ α α 1+ξ ≤ α 1+ξ C1 ψ + O √ α 2 (fα )2 C3 + + + . (83) √ α,ν,n gν ψn ν ν ν ν Estimates (81) and (83) together with (76) give the bound (0 < ξ < 1) 1+ξ
η α η α 3−ξ |(70)| ≥ 1 − O α 1+ξ C1 ψα,ν,n + O √ + + r +√ . ν ν ν ν We combine this upper bound with the lower bound (72) to arrive at √
η ν + O(n0 ) α 1+ξ η α 3−ξ 1 − O α 1+ξ C1 ψα,ν,n ≥ O + √ + + r + . α ν ν ν ν
Choose α so small that 1 − O α 1+ξ > 1/2 and take the limits η → 0, n → ∞ to get √ 1+ξ 3−ξ ν α√ C1 f g ψ = O + α ν . Take for example ξ = 1/2, ν = ν(α) = α 9/4 . α ν α + ν
Then the upper bound is O α 1/4 , so limα→0 C1 fα gν(α) ψ = 0. Since the operator C1 is semibounded its quadratic form is closed, hence fα gν(α) ψ → ψ, as α → 0, implies that ψ is in the form domain of C1 and that C1 ψ = 0. 5. Proofs of Theorems 3, 6 and of Corollary 4 In order to alleviate notation we drop in this section the variable ξ labelling the fiber (imagining ξ ∈ [ρcrit , ∞) × S 1 to be fixed). The operator Lλ,ξ , (51), is thus denoted Lλ = L0 + λ(I + K), where I and K are given in (53), (54). In parallel we can imagine that K = 0 yielding the proof of Theorem 6. Let , ρ, θ > 0 be parameters. Set Pρ = P0 P ( ≤ ρ), P0 = P (L1 = 0), A0 =
2 iθ λ(Pρ I R
2 − R I Pρ ),
R = P ρ R ,
(84) (85)
with R = (L20 + 2 )−1/2 , and where P ρ = 1l − Pρ , and we set P 0 = 1l − P0 . Define the selfadjoint operator B = C1 + i[Lλ , A0 ] = + I1 + i[Lλ , A0 ],
(86)
Stability of Equilibria with a Condensate
637
where the last commutator is a bounded operator, and decompose B = Pρ BPρ + P ρ BP ρ + 2RePρ BP ρ .
(87)
Our goal is to obtain a lower bound on Bψλ , where ψλ is a normalized eigenvector of Lλ . We look at each term in separately. In what follows we use the standard
(87) form bound λI1 ≥ − 21 − O λ2 , and the estimates 1/2 ψλ = O (λ), P 0 P ( ≤ ρ)ψλ = O (λ). The former estimate follows from Theorem 2 (or Theorem 5 for the system without condensate) and the latter is easily obtained like this: let χ ∈ C0∞ (R) be such that 0 ≤ χ ≤ 1, χ (0) = 1 and such that χ has support in a neighborhood of the origin containing no other eigenvalue of L1 than zero. Then, for ρ sufficiently small, we have P 0 P ( ≤ ρ)χ(L0 ) = 0, so P 0 P ( ≤ ρ)ψλ = P 0 P ( ≤ ρ)(χ (Lλ ) − χ (L0 ))ψλ = O (λ), by standard functional calculus. Taking into account these estimates we get
Pρ BPρ
ψλ
2 2 2 ≥ 2θλ2 Pρ I R I Pρ + θ λ2 Pρ I R KPρ + Pρ KR I Pρ − O λ2
2
≥ 2θ λ2 Pρ I R I Pρ
ψλ
ψλ
−
θ λ2
O
θ
ψλ
+ ,
(88)
where we use in the last step that P ρ = P 0 P ( ≤ ρ) + P ( > ρ) to arrive 2
at Pρ I R KPρ = Pρ I R2 P 0 P ( ≤ ρ)KPρ ≤ c. The last estimate is due to
R2 P 0 P ( ≤ ρ) < c and Pρ I P ( < ρ) < c. Next we estimate
1 2 P ρ BP ρ ψ ≥ P ρ ψ − 2θ λ2 Re P ρ (I + K)Pρ I R − O λ2 λ λ ψλ 2 2 1 λ 2 , (89)
Pρ I R = O P ρ (I + K)Pρ I R = P ρ ψλ 2 O ψλ ρ 3/2
√ where we use P ψλ ≤ P 0 P ( ≤ ρ)ψλ + P ( > ρ)ψλ = O λ/ ρ , and
ρ√
Pρ I R = O 1/ . The latter estimate is standard in this business, it follows from Pρ I R2 I Pρ = O (1/) (see e.g. [BFSS]). Estimates (89) together with P ρ ψ ≥ λ
P ( > ρ)ψλ ≥ ρ P ( > ρ)ψλ ≥ ρ( P ρ ψ − O λ2 ) gives λ
ρ θ λ2 O P ρ BP ρ ψ ≥ Pρ ψ − λ λ 2
λ2 + √ θ ρ
(90)
.
Our next task is to estimate
Pρ BP ρ
ψλ
2 = λ Pρ I1 P ρ ψ − θ λ Pρ (Lλ Pρ I R2 − I R Lλ )P ρ λ
ψλ
.
(91)
We have Pρ I1 P ρ ψ = Pρ I1 P 0 P ( ≤ ρ) ψ + Pρ I1 P ( > ρ) ψ = O (λ) + λ λ λ
O (I1 )a −1/2 1/2 ψλ = O (λ), where (I1 )a means that we take in I1 only the terms containing annihilation operators (see (53)) and where we use (I1 )a −1/2 < c.
638
M. Merkli
The second term on the r.h.s. of (91) is somewhat more difficult to estimate. We have 2 2 = −θ λ2 (I + K)Pρ I R θλ Pρ (Lλ Pρ I R2 − I R Lλ )P ρ ψ ψλ λ 2 2 2 2 −θ λ Pρ I R L0 P ρ + θ λ Pρ ((I + K)Pρ I R − I R P ρ (I + K))P ρ , ψλ
ψλ
(92)
2
where the first term on the r.h.s. comes from the contribution Pρ L0 I R ψλ in the l.h.s. by using that Pρ L0 = L0 Pρ = Lλ Pρ − λ(I + K)Pρ and that Lλ ψλ = 0. We treat the first term on the r.h.s. of (92) as 2 (I + K)Pρ I R ψλ = (I + K)Pρ I R2 P 0 P ( ≤ ρ) + (I + K)Pρ I R2 P ( > ρ) ψλ ψλ
λ = O λ P 0 P ( ≤ ρ)ψλ + O −2 (I1 )a −1/2 1/2 ψλ = O λ + 2 . (93)
In a similar way, Pρ I R2 L0 P ρ ψ = O √λ . Next we estimate the third term in the λ r.h.s. of (92) as 2 Pρ ((I + K)Pρ I R2 − I R (I + K))P ρ ψλ
λ 2 −3/2 =O
P ρ ψλ + O Pρ I R (I + K)P ρ ψλ = O √ 2 . (94) ρ Collecting the effort put into estimates (93) and (94) rewards us with the bound √ 2 Pρ BP ρ ψ = θλ O θ + + √λ ρ , which we combine with (88) and (90) to obtain λ
Bψλ ≥ 2θ λ
2
2 Pρ I R I P ρ
ρ θ λ2 O + Pρ ψ − λ ψλ 2
√ λ2 λ + √ + + √ θ ρ ρ
.(95)
2
The non-negative operator Pρ I R I Pρ has appeared in various guises in many previous papers on the subject (“level shift operator”). The result follows from a rather straightforward calculation, using the explicit form of the interaction I , (53). We do not write down the analysis, one can follow closely e.g. [BFSS, M, BFS]. Lemma 9. Let p be the parameter characterizing the infrared behaviour of the form factor (see Theorem 6; in the situation of Theorem 3 we set p = 0), 2+2p 1 ρ ρ 2 0 Pρ I R I Pρ = P0 + O( ) P0 ⊗ P ( ≤ ρ) + O (96) + 3 , 2 0 whose norm vanishes in the limit → 0, and where where O( ) is an operator = S 2 dσ |g(1, σ )|2 , and is the non-negative operator on Ran P0 which has the following matrix representation in the basis {ϕj ⊗ ϕj }dj =1 : is tridiagonal with diagonal [a, 1√+ 2a, . . . , 1 + 2a, 1 + a] and constant and equal sub- and superdiagonal with entry − a(1 + a), a = ρ(1) = eβ1−1 . The kernel of is spanned by the Gibbs state (17), and the spectrum of has a gap γ > 0 at zero which is uniform in β ≥ β0 , for β0 fixed.
Stability of Equilibria with a Condensate
639
2 2 2 It follows from the lemma that 2θ λ2 Pρ I R I Pρ ≥ 2 θλ γ P 1,β Pρ ψ − θλ O( 0 )+ λ ψ λ
2+2p O ρ + ρ2 , where P 1,β = 1l − P1,β , and P1,β = |1,β 1,β | is the projection onto the span of the Gibbs state (17). Using this estimate in (95) gives 2θ λ2 ρ 2θ λ2 Bψλ ≥ min γ,
ψλ 2 − γ P1,β P ( ≤ ρ) ψ λ 2 2 2 2+2p θλ ρ λ ρ λ (97) − O + √ + √ + O( 0 ) + + 2 . θ ρ ρ Let us choose the parameters = λ49/100 , θ = λ1/100 , ρ = λ, p > −1/2. Then the 2 −25/13 ) and the error term in minimum in (97) is given by 2θλ γ (provided λ ≤ (4γ )
1/100 (97) is O λ + O(λ0 ) = O(λ0 ). The virial theorem tells us that Bψλ = 0, so P1,β P ( ≤ λ) ψ ≥ 1 − O(λ0 ). We may write this as ψλ = P1,β P ( ≤ λ)ψλ + λ O(λ0 ) = 1,β ⊗ P ( ≤ λ)χλ + O(λ0 ), for some vector χλ ∈ F ⊗ F with norm
χλ ≥ 1 − O(λ0 ). We point out that all estimates are uniform in ξ in any compact set. This is easily seen by noticing that the only way ξ enters is through the term Kξ , which is uniformly bounded in ξ belonging to any compact set in R2 . This finishes the proofs of Theorems 3 and 6! ξ
Proof of Corollary 2. We denote by P1,β , Pβ,0 and Pβ,λ the projections onto the spans ξ
ξ
of 1,β , β,0 and β,λ , see (17), (16) and (56). Since Pβ,0 − Pβ,λ → 0 as λ → 0 (uniformly in ξ in any compact set and in β ≥ β0 , for any β0 fixed, [FM2]) it follows
ξ that ψλ = (Pβ,λ )⊥ ψλ = P β,0 ψλ + O(λ0 ) = P 1,β ⊗ P0 ψλ + P 0 ψλ + O(λ0 ) =
1,β ⊗ P 0 P ( ≤ λ)χλ + O(λ0 ), where we used (31) in the last step. It suffices now to observe that P 0 P ( ≤ λ) converges strongly to
zero, as λ → 0. This follows
from P 0 = P F ⊗ PF + 1lF ⊗ P F , P ( ≤ λ) = P (d(ω) ≤ λ) ⊗ P (d(ω) ≤ λ) P ( ≤ λ) and the fact that d(ω) has absolutely continuous spectrum covering R+ and a simple eigenvalue at zero, F being the eigenvector. Acknowledgements. I thank W. Abou Salem, J. Derezi´nski, J. Fr¨ohlich, M. Griesemer, V. Jak˘si´c, A. Joye, Y. Pautrat, C.-A. Pillet, L. Rey-Bellet, I.M. Sigal, S. Starr for interesting discussions. I am particularly grateful to J¨urg Fr¨ohlich for his patience in teaching me.
References [A] [ABG] [AW] [BFS] [BFSS]
Araki, H.: Hamiltonian formalism and the Canonical Commutation Relations in Quantum Field Theory. J. Math. Phys. 1(6), 492–504 (1960) Amrein, W., Boutet de Monvel, A., Georgescu, V.: C0 -Groups, Commutator Methods and Spectral Theory of N-body Hamiltonians. Basel-Boston-Berlin: Birkh¨auser, 1996 Araki, H., Woods, E.: Representations of the canonical commutation relations describing a non-relativistic infinite free bose gas. J. Math. Phys. 4, 637–662 (1963) Bach, V., Fr¨ohlich, J., Sigal, I.M.: Return to equilibrium. J. Math. Phys. 41(6), 3985–4060 (2000) Bach, V., Fr¨ohlich, J., Sigal, I.M., Soffer, A.: Positive Commutators and the spectrum of Pauli-Fierz hamiltonians of atoms and molecules. Commun. Math. Phys. 207(3), 557–587 (1999)
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Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I, II. Texts and Monographs in Physics. Berlin-Heidelberg-New York: Springer-Verlag, 1987 [C] Cannon, J.T.: Infinite Volume Limits of the Canonical Free Bose Gas States on the Weyl Algebra. Comm. Math. Phys. 29, 89–104 (1973) [DJ] Derezi´nski, J., Jak˘si´c, V.: Return to Equilibrium for Pauli-Fierz Systems. Ann. Henri Poincar´e 4(4), 739–793 (2003) [DJP] Derezi´nski, J., Jak˘si´c, V., Pillet, C.-A.: Perturbation theory for W ∗ -dynamics, Liouvilleans and KMS-states. Rev. Math. Phys. 15(5), 447–489 (2003) [DWRN] Diener, R.B., Wu, B., Raizen, M.G., Niu, Q.: A Quantum Tweezer for Atoms. Phys. Rev. Lett. 89, 070401 (2002) [FM1] Fr¨ohlich, J., Merkli, M.: Thermal Ionization. Math. Phys. Anal. Geom. 7(3), 239–287 (2004) [FM2] Fr¨ohlich, J., Merkli, M.: Another return of “Return to Equilibrium”. Commun. Math. Phys. 251, 235–262 (2004) [FM3] Fr¨ohlich, J., Merkli, M.: In preparation [FMS] Fr¨ohlich, J., Merkli, M., Sigal, I.M.: Ionization of atoms in a thermal field. J. Stat. Phys. 116(1/4), 311–359 (2004) [GG] Georgescu, V., G´erard, C.: On the Virial Theorem in Quantum Mechanics. Commun. Math. Phys. 208, 275–281 (1999) [H] Hugenholtz, N. M.: Quantum Mechanics of infinitely large systems. In: Fundamental Problems in Statistical Mechanics II, E.G.D. Cohen (ed.), Amsterdam: North-Holland Publishing Company New York: John Wiley & Sons, Inc., 1968 [Ha] Haag, R.: Local Quantum Physics. Tests and Monographs in Physics, Berlin-Heidelberg-New York: Springer Verlag, 1992 [HL] Hepp, K., Lieb, E.H.: Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors. Helv. Phys. Acta 46, 573–603 (1973) [JP1] Jak˘si´c, V., Pillet, C.-A.: On a Model for Quantum Friction III. Ergodic Properties of the Spin-Boson System. Commun. Math. Phys. 178, 627–651 (1996) [JP2] Jak˘si´c,V., Pillet, C.-A.:A note on eigenvalues of Liouvilleans. J. Stat. Phys. 105(5–6), 937–941 (2001) [LP] Lewis, J.T., Pul`e, J.V.: The Equilibrium States of the Free Boson Gas. Comm. Math. Phys. 36, 1–18 (1974) [LV] Lima, R., Verbeure, A.: Local perturbatoins and approach to equilibrium. Ann. Inst. Henri Poincar´e, XVII(3), 227–240 (1973) [M] Merkli, M.: Positive Commutators in Non-Equilibrium Quantum Statistical Mechanics. Commun. Math. Phys. 223, 327–362 (2001) [QV] Quaegebeur, J., Verbeure, A.: Relaxation of the Ideal Bose Gas. Lett. Math. Phys. 9, 93–101 (1985) [R] Robinson, D.W.: Return to Equilibrium. Commun. Math. Phys. 31, 171–189 (1973) Communicated by A. Kupiainen
Commun. Math. Phys. 257, 641–657 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1327-4
Communications in
Mathematical Physics
Bounded Subquotients of Pseudodifferential Operator Modules Charles H. Conley Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, TX 76203-1430, USA. E-mail: [email protected] Received: 25 May 2004 / Accepted: 30 November 2004 Published online: 22 March 2005 – © Springer-Verlag 2005
Abstract: Recently there have been several papers on the action of the Virasoro Lie algebra on the projective decompositions of the modules of pseudodifferential operators on the circle. We use their results to prove that a wide class of the uniserial (completely indecomposable) bounded modules of the Virasoro Lie algebra may be realized as subquotients of such modules of pseudodifferential operators. This gives easy proofs of the existence of many previously known uniserial modules, and moreover yields some hitherto undiscovered. 1. Introduction There are several papers concerning the indecomposable bounded representations of the Virasoro Lie algebra and its close relatives Vec(R) and Vec(S 1 ), the Lie algebras of polynomial vector fields on the line and the circle. For example, Feigin and Fuks [FF80] classified the bounded representations of Vec(R) of length 2 (as a corollary of deep cohomological results), Martin and Piard [MP92] classified the bounded representations of the Virasoro Lie algebra of weight space dimensions all less than or equal to 2, and the author [Co01] classified the regular (see below) bounded representations of Vec(S 1 ) of length 3. In all of these works the strategy is to compute the 1-cocycles and cup products associated to the irreducible bounded representations, the tensor density modules. This leads to difficult calculations, particularly in the positive cases where one must prove that indecomposable representations of a given composition series do exist. (The calculations are easier in the negative cases of composition series which do not admit indecomposable representations.) Here we take a different approach to the positive cases: we prove that uniserial (i.e., completely indecomposable) bounded representations with certain composition series
Partially supported by NSA grant MDA 904-03-1-0004.
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do exist, by realizing them as subquotients of modules of pseudodifferential operators. This has the advantage of yielding essentially all previously known positive results, in addition to several new ones, relatively simply as corollaries of the work of Cohen, Manin, and Zagier [CMZ97] on projective decompositions of pseudodifferential operator modules in the regular case, and the work of the author and Sepanski [CS04] and Gargoubi [Ga00] in the singular case. It has the disadvantage of yielding no information on the negative cases, but as we mentioned, the direct approach is not so difficult there. In fact, it can be carried out with a computer; see Sect. 7. Let us describe some of the salient points in more detail. Without precisely specifying the allowed class of functions, define A(γ ) := {f (z)dzγ : f a function on S 1 }, the space of tensor densities of degree γ . This space is naturally a Vec(S 1 )-module, and restriction to R yields a Vec(R)-module which we also denote by A(γ ). It was proven in [FF80] that there exist indecomposable Vec(R)-modules of length 2 with composition series (A(γ ), A(γ + p)) under the following conditions: for √ all γ when p = 2, 3, or 4, only for γ = −4 or 0 when p = 5, only for γ = (−5 ± 19)/2 when p = 6, and for no γ when p ≥ 7. One of the results of [MP92] is that exactly the same thing is true for Vec(S 1 )-modules. (Such modules also exist for p = −1, 0, and 1, but, excepting the case with composition series (A(0), A(1)), we will not discuss them because they are not realized as subquotients of pseudodifferential operator modules.) In [Co01] the author considered uniserial Vec(S 1 )-modules of length 3 with composition series (A(γ ), A(γ + p), A(γ + p + q)), regular in the sense that the eigenvalues of the Casimir operator of the subalgebra of projective transformations on the composition series modules are distinct. He proved that such modules exist for generic γ when p + q < 7, for exceptional γ when 7 ≤ p + q < 9, and for no γ when p + q ≥ 9. √In particular, he proved that when p = q = 4, such modules exist only for γ = (−7± 39)/2. (The cutoffs 5, 7, and 9 in the length 2 and length 3 cases are explained by the structure of 2 Vec(R) as a projective module.) All of these results require substantial calculations, especially the length 2 cases with p = 6 and the length 3 cases with p = q = 4. In the present paper we will see that the existence of all these indecomposable modules follows easily from [CMZ97], which can be used to prove that they are realized as subquotients of pseudodifferential operator modules. Moreover, the same method proves the existence of various new uniserial modules, including several singular modules. For example, for each k ∈ 3 + N there exists a uniserial module with composition series (A(γ ), A(γ + 4), A(γ + 6), . . . , A(γ + 2k − 6), A(γ + 2k − 4), A(γ + 2k)) when γ = (1−2k ± (2k − 2)2 + 3)/2. At k = 3 and 4 this yields the cases of length 2, p = 6 and length 3, p = q = 4 mentioned above. Our results remain essentially unchanged if Vec(S 1 ) is replaced by Vec(R) (see Sect. 7). It would be interesting to use our strategy to produce uniserial representations of Vec(Rn ) composed of tensor density modules. Some of the necessary formulae for projective decompositions of pseudodifferential operator modules may be found in [LO99]. This paper is organized as follows. In Sects. 2 and 3 we establish notation and state our main theorem. In Sects. 4 and 5 we recall the necessary results concerning pseudodifferential operators, in Sect. 6 we prove our main theorem, and in Sect. 7 we make some miscellaneous remarks.
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2. Background Throughout this paper we work in the algebraic setting. We expect that our main results hold also in the smooth setting, with essentially the same proofs given here. Let us write D for the differentiation operator d/dz. The complex Lie algebra Vec(S 1 ) of polynomial vector fields on the circle, the centerless Virasoro Lie algebra, has basis {zn D : n ∈ Z} and the usual Lie bracket of vector fields. Given a representation W of Vec(S 1 ), we write Wµ for the µ-eigenspace of zD, the µ-weight space. Representations such that zD acts semisimply with weight spaces of uniformly bounded dimension are said to be bounded. The irreducible bounded representations have been classified [MP91, Ma92]; all are tensor density modules. For any a, γ in C, the tensor density module A(a, γ ) has basis {dzγ zλ−γ : λ ∈ a+Z}. (This is the notation used in [MP92] and [Co01]. In [FF80], A(a, γ ) is called F−γ ,a−γ .) The Vec(S 1 ) action is πa,γ (zn+1 D)(dzγ zλ−γ ) = (λ + nγ )dzγ zλ+n−γ . In particular, zD has spectrum a+Z, and the weight space A(a, γ )λ is the line Cdzγ zλ−γ . It is simple to verify the following properties of the A(a, γ ). The only equivalences between them are A(a, 0) ∼ = A(a, 1) for a = 0, and the (restricted) dual of A(a, γ ) is A(−a, 1 − γ ). The only reducible A(a, γ ) are A(0, 0) and its dual A(0, 1); the 0weight space of A(0, 0) is the trivial representation, which we denote by D0 , and the quotient representation A˜ := A(0, 0)/D0 is irreducible. Therefore the classification of ˜ A(a, 0) for a = 0, irreducible bounded representations of Vec(S 1 ) is as follows: D0 , A, and A(a, γ ) for γ = 0 or 1, where 0 ≤ Re(a) < 1. It follows immediately that any bounded representation of Vec(S 1 ) is of finite length, with a composition series of tensor density modules. It is also clear that in any indecomposable bounded representation, the spectrum of zD is a single coset of Z in C. Thus for example a representation with composition series (A(a1 , γ1 ), . . . A(an , γn )) can be indecomposable only if a1 = · · · = an . Suppose now that g is any Lie algebra, and for 1 ≤ i ≤ n, ψi is a representation of g on a space Vi . We define an extension of V1 → V2 → · · · → Vn to be a representation ψ of g on a space W , such that W admits a ψ-invariant flag W = W1 ⊃ W2 ⊃ · · · ⊃ Wn+1 = 0 whose subquotient representations Wi /Wi+1 are equivalent to Vi for all i. In particular, if the ψi are irreducible then ψ is a representation of length n with Jordan-H¨older composition series (ψ1 , . . . , ψn ). Note that the dual of an extension of V1 → · · · → Vn is an extension of Vn∗ → · · · → V1∗ . We will be concerned with the case of extensions (ψ, W ) of V1 → · · · → Vn such that the invariant flag {Wi } splits in the category of vector spaces, i.e., W has subspaces F1 , . . . , Fn such that Wi = ⊕nj=i Fj . Here W = ⊕n1 Fj , and so we may regard any endomorphism T : W → W as an n × n matrix with entries Tij : Fj → Fi . In particular, we may regard the representation ψ : g → End(W ) as a matrix with entries ψij : g → Hom(Fj , Fi ). Let us examine this matrix. The fact that ψ preserves ⊕nj=i Fj for all i implies that ψij = 0 for i < j , and the fact that the subquotient representation of ψ on Wi /Wi+1 is equivalent to ψi implies that ψii is a representation of g on Fi , equivalent to ψi . The fact that ψ is a representation can be rephrased in cohomological terms as the cup equation:
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∂ψij +
ψik ∪ ψkj = 0 for all i > j.
i>k>j
Here we regard ψij as a Hom(Fj , Fi )-valued 1-cochain of g, and use the standard definitions of the coboundary operator ∂ and the cup product ∪ for Lie algebra cohomology (see for example [Co01] for a brief summary). Note that the entries ψj +1,j on the first subdiagonal of the matrix of ψ are cocycles. A representation is said to be uniserial, or completely indecomposable, if it admits only one flag of subrepresentations with irreducible subquotients, or equivalently, if all of its subquotients are indecomposable. For example, A(0, 0) is a uniserial extension of ˜ Note that subquotients A˜ → D0 , and its dual A(0, 1) is a uniserial extension of D0 → A. and duals of uniserial representations are uniserial. In this paper we only discuss uniserial bounded representations, as it is conceivable that these could be classified. It is not possible to classify all bounded representations; Germoni has shown that this is a wild problem [Ge01]. The following lemma is elementary. Lemma 2.1. Let ψ be an extension of V1 → · · · → Vn with invariant flag ⊕ni Fj and matrix entries ψij as above, and assume in addition that the subquotient representations Vi are all irreducible. Then ψ is uniserial if and only if the cocycles ψj +1,j are all non-trivial. 3. Main Results In this section we list various composition series which admit uniserial extensions, most of which arise as subquotients of modules of pseudodifferential operators. The proofs are given in Sect. 6. We only prove the existence of these uniserial representations; our method does not classify the uniserial representations of any given composition series. See Sect. 7 for an approach to the classification problem using computers. The length 2 and regular length 3 uniserial bounded representations have already been classified [FF80, MP92, Co01]. In these cases the new aspect of our results is that the representations are realized as subquotients of pseudodifferential operator modules. In the other cases the extensions themselves were not previously known. Definition. We shall refer to an extension of A(a, γ ) → A(a, γ + p1 ) → · · · → A(a, γ + p1 + · · · + pk ) as a jump (p1 , . . . , pk ) extension with parameters (a, γ ). Note that a jump (p1 , . . . , pk ) extension with parameters (a, γ ) is of length k + 1, unless a = 0 and 0 or 1 occurs among the γ + p1 + · · · + pi ; its length increases by 1 for each such occurrence. Also, its dual is a jump (pk , . . . , p1 ) extension with parameters (−a, 1 − pk − · · · − p1 − γ ). Let us emphasize that in the cases of the following theorem in which we obtain uniserial modules for almost all values of the parameter γ , we usually obtain the singular uniserial modules in addition to the regular ones. At the outset this phenonmenon is surprising, as the methods of [MP92] are completely different in the regular and singular cases, and those of [Co01] break down in the singular case. However, it was predicted and explained in Sect. 9.3 of [CS04], which extends parts of [CMZ97] to the singular case. Nevertheless, to our knowledge the only singular uniserial modules explicitly known prior to the present paper were of length 2 [FF80, MP92].
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Perhaps the most interesting extensions below are those which were previously unknown and involve special values of γ : see parts (c)–(d) and (h)–(j), as well as the singular extensions of length exceeding 2, particularly those in part (k). Theorem 3.1. All of the following classes of representations contain uniserial representations. Moreover, all of these uniserial representations are realized as subquotients of modules of pseudodifferential operators, with the possible exception of those in part (d). (a) The jump (2, 2, . . . , 2) extensions of any length, for any parameters (a, γ ). In fact, for each length and value of (a, γ ) there is a 1-parameter family of such extensions. For generic values of the parameter the extension is uniserial, and for γ ∈ −1/2−N it is singular whenever it is long enough. (b) The jump (4, 2, 2, . . . , 2) and (2, 2, . . . , 2, 4) extensions of any length r ∈ 2 + N, for any parameters (a, γ ), with the following exceptions. First, γ cannot be −3, √ −1/2, 0, or 1. Second, if 3γ = −k − 1 ± √(k + 1)(k − 2) for some k ∈ N, then 2r cannot exceed k. Third, if 2γ = −2k−1± 4k 2 + 3 for some k ∈ 2+N, then r cannot exceed k. These modules are singular for γ ∈ −3/2 − N and γ + 2r ∈ 5/2 + N. (c) The jump (4, 2, 2, . . . , 2, 4) extensions with parameters (a, γ ) and length k, where √ 2 a is arbitrary, k ∈ 2 + N, and 2γ = −2k − 1 ± 4k + 3. In the case k = 2 these should be interpreted as jump 6 extensions; here the uniserial subquotients realize the representations of length 2 discovered in [FF80] and [MP92]. √ (d) The jump (2, 4, 2) extensions with a arbitrary, γ = (−7 ± 31)/2. (e) The jump (3, 4) extensions with a arbitary, γ = −13/2, and the dual jump (4, 3) extensions with γ = 1/2. (f) The jump (3, 3) extensions for generic values of the parameters (a, γ ), including the singular extensions at γ = −5/2, −1, and −4. (g) The jump (3, 2, 2) extensions for generic values of the parameters (a, γ ), including the singular extensions at γ = −11/2, −9/2, −7/2, −3, and −1, and the dual jump (2, 2, 3) extensions. Note that these representations have uniserial jump (3, 2) and (2, 3) extensions as subquotients. (h) The jump (3, 2, 3) extensions with a arbitary, γ = −31/4 or √ 3/4. (i) The jump (3, 3, 2) extensions with a arbitary, √γ = (−27 ± 649)/8, and the dual jump (2, 3, 3) extensions with γ = (−29 ± 649)/8. √ (j) The jump (3, 2, 2, 2) extensions with a arbitary, γ =√(−67 ± 3529)/16, and the dual jump (2, 2, 2, 3) extensions with γ = (−61 ± 3529)/16. (k) The following extensions containing the jump 1 extension A(a, 0) → A(a, 1) as a subquotient, for any a ∈ C (including a = 0): the jump (1, 2, 3), jump (1, 3, 2), jump (1, 2, 2, 2), jump (2, 1, 3), jump (2, 1, 2, 2), jump (2, 1, 4), and jump (3, 1, 2, 2) extensions, as well as their duals and subquotients. 4. Pseudodifferential Operators We now define the Vec(S 1 )-modules of differential operators and pseudodifferential operators. It will be convenient to unify our treatment of the tensor density modules A(a, γ ) as follows. Define A(γ ) = A(a, γ ) = Span{dzγ zλ : λ ∈ C}. 0≤Re(a)<1
Note that A(0) is the span of the monomial functions (possibly multivalued) on S 1 , A(γ ) is the space of all products dzγ f (z) with f ∈ A(0), and Vec(S 1 ) is the space of all
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vector fields g(z)D with g ∈ A(0, 0). Let πγ be the direct sum action ⊕0≤Re(a)<1 πa,γ of Vec(S 1 ) on A(γ ). It may be written concisely as follows: for any f ∈ A(0) and g ∈ A(0, 0), πγ (gD)(dzγ f ) = dzγ (gf + γ g f ). For any p ∈ C, j ∈ N, and h(z) ∈ A(0), the differential operator dzp h(z)πγ (D)j from A(γ ) to A(γ + p) is defined by dzp h(z)πγ (D)j (dzγ g) = dzγ +p hg (j ) , where g (j ) denotes the ordinary j th -derivative of g. Let E k (γ , p) be the space of all differential operators from A(γ ) to A(γ + p) of order ≤ k: E k (γ , p) = Span{dzp h(z)πγ (D)j : h ∈ A(0), j = 0, 1, . . . , k}. It carries the natural 2-sided Vec(S 1 ) action σγ ,p , defined by σγ ,p (X)(L) = πγ +p (X) ◦ L − L ◦ πγ (X) for all X ∈ Vec(S 1 ) and L ∈ E k (γ , p). We often write simply D for πγ (D), E k for E k (γ , p), and so on when γ and p are clear from the context. For any g ∈ A(0, 0) and h ∈ A(0), a calculation yields the well known formula σγ ,p (gD)(dzp hD j ) ∞ j j − i (i+1) j −i = dzp gh + (p − j )g h D j − h γ+ . (1) D g i i+1 i=1
j (In fact, the sum 1 is really only a sum 1 , as the generalized binomial coeffij cients i = j (j − 1) · · · (j − i + 1)/ i!, defined for all j ∈ C and i ∈ N, are zero
for j ∈ N and j < i. We have written ∞ 1 for compatibility with pseudodifferential operators.) A pseudodifferential operator ( DO) of order j ∈ C from A(γ ) to A(γ + p) is a
formal sum i∈N dzp hj −i (z)πγ (D)j −i , where each hj −i is in A(0) and hj = 0. For any k ∈ C, let k (γ , p) be the space of all DOs from A(γ ) to A(γ + p) of order in k − N: k (γ , p) = dzp hk−i πγ (D)k−i : hk−i ∈ A(0) ∀ i ∈ N .
∞
i∈N
For k ∈ N, k contains the space E k of differential operators. For general k the elements of k are not well defined operators on A(γ ), but nevertheless Eq. 1 extends the action σγ ,p of Vec(S 1 ) to them. The next lemma is an immediate consequence of said equation. Lemma 4.1. The subquotient representation of σγ ,p on k / k−1 is equivalent to A(p− k) by the isomorphism dzp h(z)D k → dzp−k h(z). Therefore, for any j ∈ C and i ∈ j + N the subquotient representation of σγ ,p on p−j / p−i−1 is an extension of A(j ) → · · · → A(i), with invariant flag { p−n / p−i−1 : j ≤ n ≤ i}. This flag is split in the category of vector spaces.
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Recall that A(γ ) and A(1 − γ ) are dual. In fact, they are paired by a non-degenerate Vec(S 1 )-invariant bilinear form which is essentially the residue map. As is discussed from slightly different points of view in [CMZ97, LO99 and CS04], this generalizes to an invariant pairing of (γ , p) with (γ + p, −p), known as the Adler trace or the noncommutative residue. One obtains the following lemma. Lemma 4.2. The category of bounded subquotients of DO modules is closed under duals: ∗ p−j (γ , p)/ p−i−1 (γ , p) ∼ = i−1−p (γ + p, −p)/ j −2−p (γ + p, −p).
5. Projective Decompositions The extensions in Lemma 4.1 are almost never uniserial. In order to pick out their uniserial subquotients we must recall some recent results on their decomposition under the projective Lie algebra a, the infinitesimal linear fractional transformations: a = Span{D, zD, z2 D}. This is a subalgebra of Vec(S 1 ) isomorphic to sl2 . Let Q be its Casimir operator, the element (zD)2 − zD − (z2 D)D of its universal enveloping algebra (this is not to be confused with its image as a differential operator, which is zero). One checks that Q acts by the scalar πγ (Q) = γ 2 − γ on A(γ ). Definition. An extension either of A(a, γ1 ) → · · · → A(a, γn ) or of A(γ1 ) → · · · → A(γn ) is regular if the eigenvalues γi2 − γi of Q on the elements of its composition series are all distinct, and singular otherwise. Thus the extension is regular if and only if γi is neither γj nor 1 − γj whenever i = j . Regular Subquotients. One checks that the extension (σγ ,p , p−j / p−i−1 ) of A(j ) → · · · → A(i) is singular if and only if −2j ∈ N and i ≥ 1. In the regular case, its invariant flag splits under the action of a along the eigenspaces of σγ ,p (Q). This leads to the following analysis, all of which is implicit in [CMZ97] and explicit in [Co01]. There is a specific equivalence, which we will refer to as the projective equivalence, from (σγ ,p , p−j / p−i−1 ) to a representation π(γ , p) of Vec(S 1 ) on ⊕im=j A(m). We regard π(γ , p) as a matrix with entries πmn (γ , p) : Vec(S 1 ) → Hom(A(n), A(m)) (keep in mind that m and n are in j + N, where j is arbitary in C). This matrix has the following properties. The flag {⊕im=n A(m) : j ≤ n ≤ i} is invariant, so π(γ , p) is a lower triangular matrix. Its diagonal entries are πmm = πm , the natural action of Vec(S 1 ) on A(m). The entries πn+1,n on the first subdiagonal are all zero. The lower subdiagonal entries πmn with m ≥ n + 2 are zero on a and are a-covariant maps, i.e., they are a-relative 1-cochains. They are given explicitly by the formula πmn (γ , p) = bmn (γ , p)βm−n (n),
(2)
where the bmn are scalars depending on γ and p and βm−n (n) is a certain differential operator-valued 1-cochain independent of γ and p. In this paper we will be concerned primarily with the bmn , but we give the definition of βm−n (n) for completeness.
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Definition. For any ν ∈ C and q ∈ 2 + N, βq (ν) is the unique a-relative E q−2 (ν, q)valued 1-cochain of Vec(S 1 ) such that βq (ν)(z3 D) = 6dzq D q−2 ,
βq (ν)(z−1 D) = −6dzq z−2q (z2 D)q−2 .
In order to decide questions of uniseriality we need the following lemma. Its proof is elementary and follows for example from Lemma 3.1 of [Co01]. Note that for any a ∈ C, βq (ν) maps the subrepresentation A(a, ν) of A(ν) into A(a, ν + q). It will be convenient to define c(ν, q) = 2ν + q − 1. Lemma 5.1. Assume that ν ∈ C, q ∈ 2 + N, and c(ν, q) = 0. Then whenever βq (ν) is a cocycle, it restricts to a non-trivial Hom(A(a, ν), A(a, ν + q))-valued cocycle for all a ∈ C. The crucial result is the formula for the bmn (γ , p), which as we mentioned is implicit in [CMZ97] and explicit in[Co01]. It is best written in terms of certain auxiliary scalars , defined by their relation to the b : bmn mn 2m − 2 −1 p − n (−1)m−n−1 m − n b (γ , p). (3) bmn (γ , p) = m − n mn 6 2 m−n−2 (γ , p) = Writing c for c(γ , p), the formula is bmn (m − n)2 − 1 m − n m + n − 3 −1 c − n 2 m−n m+n−1 2 m−n−2 m−n−2 −1 m − n + 1 2n − 1 + l c−n − l l l l=0 × (m − n − l − 1)p − (m − n − l + 1)c + (m + n + l − 1) .
(4)
Singular Subquotients. Suppose now that −2j ∈ N and i ≥ 1, so that the extension (σγ ,p , p−j / p−i−1 ) is singular. This means that either 0, 1/2, or 1 appears in the interior of the sequence j, j + 1, . . . , i. Here some of the eigenvalues of the Casimir operator Q on the composition series {A(m) : j ≤ m ≤ i} are double, and in general the invariant flag is not a-split. This situation was analyzed in [Ga00 and CS04]. (However, there are errors in [Ga00]. The formulae given there for the objects defined below are only correct for the scalars a1−n,n and the restriction of the 1-cochain αq of Vec(S 1 ) to the projective subalgebra a.) We now summarize the necessary results. Just as in the regular case, there is an equivalence from (σγ ,p , p−j / p−i−1 ) to a certain representation π(γ , p) of Vec(S 1 ) on ⊕im=j A(m). As before, we regard π as a lower triangular matrix with entries πmn : A(n) → A(m), and we continue to refer to the equivalence as the projective equivalence, despite the fact that the matrix π is no longer diagonal on a, because it is as close to diagonal on a as possible. The flag {⊕im=n A(m) : j ≤ n ≤ i} is still invariant, and the diagonal entries are still πmm = πm , but the subdiagonal entries πmn with m > n are now divided into three types: regular, antidiagonal, and singular. Let us first state all the formulae together and then define their terms.
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The regular entries are those with either m < 1, or n > 0, or m+n = 0, or m+n = 2. They are given by the same formula that gives the πmn in the regular case: the regular entries πn+1,n on the first subdiagonal are all zero (note that these do not include π1,0 , which is antidiagonal), and those with m ≥ n + 2 are given by Eq. 2. The antidiagonal entries are those with m + n = 1. They are π1−n,n (γ , p) = a1−n,n (γ , p)α1−2n (n). The singular entries are those with either m ≥ 1 and m + n < 0, or n ≤ 0 and m + n > 2. For m ≥ 1 and m + n < 0, πmn (γ , p) = bmn (γ , p)βm−n (n) + am,1−m (γ , p)b1−m,n (γ , p)δ2m−1,m−n (n), while for n ≤ 0 and m + n > 2, πmn (γ , p) = bmn (γ , p)βm−n (n) + bm,1−n (γ , p)a1−n,n (γ , p)δm−n,1−2n (n). In all of these formulae the b’s and β’s are as in the regular case, the a’s and b’s are new scalars, and the α’s and δ’s are new differential operator-valued 1-cochains. The a’s are given by 1 p−n c−n (5) (1 − 2n)2 . a1−n,n (γ , p) = − 2 1 − 2n 1 − 2n We will omit the formula for the bmn , as it is so complicated (see [CS04]) that it does not seem worthwhile to treat cases involving it. Just as for the β’s, we only need the most elementary properties of the α’s and δ’s, but we give their definitions for completeness. First we must define the affine Bol operator. Definition. For any ν ∈ C and q ∈ N, the affine Bol operator Bolq (ν) is the element dzq D q of E q (ν, q). The α’s and δ’s are defined in terms of the 1-cochain βq , the operator Bolq regarded as a 0-cochain, its coboundary ∂ Bolq , and the scalar b2,0 from Eq. 3 (in the definition of α1 , take β1 to be zero). We remark that ∂ Bolq is E q−1 -valued, and Bolq itself is invariant under the affine subalgebra b = Span{D, zD} of a. It follows that the α’s and δ’s are b-relative. Definition. For any ν ∈ C and q ∈ 1 + N, including those with c(ν, q) = 0, αq (ν) is the E q−1 (ν, q)-valued 1-cochain of Vec(S 1 ) defined by the continuous extension of the formula αq (ν) = 2 b2,0 (ν, q)βq (ν) − ∂ Bolq (ν) /qc(ν, q). Definition. For any ν ∈ C, q ∈ 1 + N, and r ∈ 2 + N, including those with c(ν, q) or c(ν + r, q) equal to zero, δq+r,q (ν) and δq,q+r (ν) are the E q+r−3 (ν, q + r)-valued 1-cochains of Vec(S 1 ) defined by the continuous extensions of the formulae δq+r,q (ν) = −2 βr (ν + q) ◦ Bolq (ν) − βq+r (ν) /qc(ν, q), δq,q+r (ν) = 2 Bolq (ν + r) ◦ βr (ν) − βq+r (ν) /qc(ν + r, q).
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It is a key point (proven in Sect. 7 of [CS04]) that the formulae in these two definitions have removable singularities when their denominators are zero, as all the α’s and δ’s occurring among the entries of π are of this type. The following analog of Lemma 5.1 is needed to decide uniseriality questions. It is elementary; proofs may be found in both [Ga00] and [CS04]. Note that for any a ∈ C, αq (ν) maps A(a, ν) into A(a, ν + q). Lemma 5.2. Assume that ν ∈ C, q ∈ 1 + N, and c(ν, q) = 0. Then αq (ν)|a restricts to a non-trivial Hom(A(a, ν), A(a, ν + q))-valued 1-cocycle of a for all a ∈ C. In particular, whenever αq (ν) is a cocycle of all of Vec(S 1 ), it is non-trivial. Special Subquotients. The fact that A(γ ) and A(1 − γ ) are dual yields a Vec(S 1 )-equivalence from k (γ , p) to k (1 − p − γ , p), which is in fact conjugation of differential operators. It is of order 2, and so when 1−p −γ = γ , i.e., c(γ , p) = 0, it is an involution 1−p of k (γ , p). Here γ = 1−p 2 and the domain and range tensor density modules A( 2 ) and A( 1+p 2 ) are dual. In this case we say that (γ , p) is special. Special subquotients (σγ ,p , p−j / p−i−1 ) can be either regular or singular. They decompose into the ±1-eigenspaces of the involution, leading to the next two lemmae, which are very useful in constructing uniserial extensions all of whose jumps are even. Lemma 5.3 was proven in [CMZ97] and [CS04] in the regular and singular cases, respectively, and Lemma 5.4 was proven in Sect. 8 of [CS04]. , b , and a Lemma 5.3. For fixed m, n, and p, the scalars bmn , bmn mn mn are polynomim−n als in c(γ , p) of parity (−1) . In particular, they are zero when c(γ , p) = 0 and m − n is odd. Therefore, in the special case the matrix entries πmn (γ , p) are zero for m − n odd, and so the representation π(γ , p) of Vec(S 1 ) on ⊕im=j A(m) decomposes into ⊕m−j even A(m) and ⊕m−j odd A(m).
Lemma 5.4. Suppose that (σγ ,p , p−j / p−i−1 ) is both singular and special, and in addition that −j ∈ N. Then all of the associated scalars a1−n,n are zero, and all of the associated bmn are given by the formula for bmn , with all terms containing a factor of c deleted (some of these terms will have zeroes in their denominators, but the remaining undeleted terms will not). 6. Proofs We begin by defining various useful Vec(S 1 )-subrepresentations of the DO modules k (γ , p). Recall from the last section that for i − j ∈ N, we have the projective equivalence from p−j / p−i−1 to the representation π(γ , p) of Vec(S 1 ) on ⊕ij A(m). (We did not actually define the projective equivalence; that is done in [CMZ97] in the regular case and [CS04] in the singular case.) For any j ∈ C and 0 < k1 < · · · < kr ∈ N, let us write A(j ; k1 , . . . , kr ) for the r +1 subspace of ⊕kl=0 A(j + l) obtained by omitting the summands with l = ki , 1 ≤ i ≤ r. p−j
Definition. For any γ , p, and j in C and 0 < k1 < · · · < kr ∈ N, let k1 ,...,kr (γ , p) be the subspace of p−j (γ , p) which maps to A(j ; k1 , . . . , kr ) under the composition of the quotient map from p−j to p−j / p−j −kr −2 with the projective equivalence from j +k +1 this quotient to ⊕m=jr A(m).
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p−j
Clearly k1 ,... ,kr (γ , p) is a Vec(S 1 )-subrepresentation of p−j (γ , p) if and only if j +k +1
A(j ; k1 , . . . , kr ) is a π(γ , p)-subrepresentation of ⊕j r A(m). Therefore the next lemma follows easily from the fact that π(γ , p) is a lower triangular matrix, together with Lemma 5.3. We leave the details to the reader. Lemma 6.1. Fix γ , p, j ∈ C, and write k for k (γ , p), π for π(γ , p), etc. p−j
(a) 1,2,... ,r is Vec(S 1 )-invariant whenever the matrix entries πj +1,j , πj +2,j , . . . , πj +r,j are all zero. p−j (b) 1,3 is Vec(S 1 )-invariant whenever πj +1,j , πj +3,j , and πj +3,j +2 are all zero. p−j
(c) 1,2,4 is Vec(S 1 )-invariant whenever πj +1,j , πj +2,j , πj +4,j , and πj +4,j +3 are all zero. p−j (d) In the special case, 1,3,5,... ,2r−1 is Vec(S 1 )-invariant for all r ∈ N. p−j
(e) In the special case, 1,2,3,5,7,9,... ,2r−1 is Vec(S 1 )-invariant for all r ∈ N whenever πj +2,j = 0. In order to use Lemma 6.1 we must analyze the scalars bmn . Recall that for p −j ∈ N, p−j decomposes as E p−j ⊕ −1 under Vec(S 1 ). Therefore in this case the matrix of π(γ , p) is block diagonal: πmn = 0 whenever p − m ∈ N and p − n ∈ Z− . Equation 3 . In other shows that for the regular entries, this is effected by the quotient bmn /bmn words, we have the following obvious lemma. Lemma 6.2. Suppose that m − n ∈ 2 + N and πmn is of the form bmn βm−n . If p − n ∈ N and p − m ∈ Z− , then bmn = 0 regardless of whether or not bmn = 0. Conversely, − if bmn = 0 but bmn = 0, then p − n ∈ N and p − m ∈ Z . . The next three equations are simpliLemma 6.2 permits us to focus on the bmn fications of Eq. 4 at m − n = 2, 3, 4 (we suppress the argument (γ , p) of bmn and c): (2n + 1)bn+2,n = 3c2 − (n2 + n + 1) − p(2n + 1), n(n + 1)bn+3,n = 2c c2 − 1 − p(n + 1) , 2n(2n + 1)(2n + 3)bn+4,n = 15c4 + 5c2 2n2 + 6n − 3 − 2p(2n + 3) −n(n + 3) n2 + 3n + 6 + 2p(2n + 3) .
(6) (7)
(8)
To obtain uniserial representations of the polynomial version of Vec(S 1 ) we are using, we need the subrepresentations of k analogous to the subrepresentations A(a, γ ) of A(γ ). Note that the monomial DO dzp zµ D k is of weight µ − p + k under σγ ,p (zD). Therefore, for a ∈ C we define k (a, γ , p) = dzp hk−i πγ (D)k−i : hk−i ∈ A(a − p + k, 0) ∀ i ∈ N , i∈N
the subrepresentation of k (γ , p) with weights in a + Z. Henceforth we will often use p−j as an abbreviation for p−j (a, γ , p) rather than p−j (γ , p); this should not cause confusion. For i − j ∈ N, it is clear that p−j / p−i−1 is an extension of p−j A(a, j ) → · · · → A(a, i). We define k1 ,... ,kr (a, γ , p) similarly, and henceforth use p−j
k1 ,... ,kr as an abbreviation for it.
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Proof of Theorem 3.1. First, let us explain why we do not need to treat the case that A(0, 0) or A(0, 1) is in the composition series separately. It will suffice to discuss A(0, 0), as A(0, 1) is its dual. We have mentioned that A(0, 0) is itself a uniserial extension of A˜ → D0 . We know from [MP92] that all of the following subquotients are uniserial: A(0, k) → A(0, 0) for k = −4, −3, or −2 and b0,k = 0, and A(0, 0) → A(0, k) for either k = 1 and a1,0 = 0 or k = 2 and b2,0 = 0. On the other hand, the subquotients A(0, 0) → A(0, k) for k = 3, 4, and 5 and bk,0 = 0 are not uniserial; they have uniserial subquotients A˜ → A(0, k). However, we are not discussing cases involving b coefficients. For parts (a) through (d) we will use the special case c = 0, where by Lemma 6.1 we p−j p−j have the representation 1,3,... ,2r−1 , and sometimes also 1,2,3,5,... ,2r−1 . Part (a). For c = 0, 1,3,... ,2r−1 / p−j −2r−1 is a jump (2, . . . ,2) (r jumps) extension with parameters (a, j ). Combining Lemmas 2.1, 5.1, and 5.2, we see that it is uniserial if and only if all of the matrix entries πj +2,j , πj +4,j +2 , . . . , πj +2r,j +2r−2 are non-zero. Now all of these entries are regular, except when the antidiagonal entry π3/2,−1/2 occurs. For any given j ∈ C, Eqs. 5 and 6 and Lemma 6.2 show that they are all non-zero for almost all values of the free parameter γ , proving part (a). p−j
Part (b). We shall discuss only the jump (4, 2, . . . , 2) extensions, as the jump (2, . . . , p−j 2, 4) extensions are their duals. Whenever c = 0 and πj +2,j = 0, 1,2,3,5,... ,2r−1 / p−j −2r+1 is a jump (4, 2, . . . , 2) extension of length r with parameters (a, j ). Just as in part (a), Lemmas 2.1, 5.1, and 5.2 show that it is uniserial if and only if πj +4,j , πj +6,j +4 , . . . , πj +2r,j +2r−2 are all non-zero. This is the case for generic values of j , but there are various exceptions. Let us consider first the case that all of the entries which must be non-zero are given by Eq. 2. By the definition of the regular entries in the singular case together with Lemma 5.4, this occurs unless j = −3/2 or j ∈ −1/2 − 2N. Here πj +2,j = 0 implies bj +2,j = 0, and πj +4,j = 0 implies bj +4,j = 0. By Lemma 6.2, this can only occur if bj +2,j = 0. Since we are in the special case c = 0, Eq. 6 yields p = −(j 2 + j + 1)/(2j + 1). The exceptions in the statement of the theorem arise as follows. First, uniseriality fails if bj +4,j = 0. At c = 0, Eq. 8 becomes = −(n + 3) n2 + 3n + 6 + 2p(2n + 3) . 2(2n + 1)(2n + 3)bn+4,n
(9)
Solving the system bj +4,j = bj +2,j = c = 0 gives j = −3, −5/2, 0, or 1. At j = −5/2, π3/2,−5/2 is not a regular entry, so we are not yet considering this case. Second, if p −j = k ∈ 4+N, then Lemma 6.2 shows that the quotient bj +2r,j +2r−2 / bj +2r,j +2r−2 is zero for 2r = k + 1 or k + 2, so we can only have uniseriality for 2r ≤ k. Similarly, if p − j = 0, 1, 2, or 3, then bj +4,j itself is zero. To see when these situations arise, we solve p − j = k and c = bj +2,j = 0 simultaneously to obtain √ 3j 2 + 2(k + 1)j + (k + 1) = 0, or 3j = −k − 1 ± (k + 1)(k − 2). Third, if bj +2k+2,j +2k = 0 for some k ∈ 2 + N, then we can only have uniseriality for r ≤ k. At c = 0, one checks that bj +2,j = bj +2k+2,j +2k = 0 if and only if √ √ 2j = −2k − 1 ± 4k 2 + 3 and 2p = ∓ 4k 2 + 3 (see Eq. 10). Now consider the cases in which some of the entries that must be non-zero are not given by Eq. 2. First, if j ∈ −9/2 − 2N and j + 2r ∈ 3/2 + N, then one of these entries
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is π3/2,−1/2 = a3/2,−1/2 α2 (−1/2). The remaining entries are regular, so by the earlier part of the proof we need only verify that a3/2,−1/2 = 0 at c = bj +2,j = 0. By Eq. 5, if c = 0 then a3/2,−1/2 = 0 if and only if p = ±1/2. The result now follows from (2j + 1)p + j 2 + j + 1 = 0. Next, suppose that j = −3/2. Here by the earlier part of the proof we only need to check that a5/2,−3/2 = 0 when c = 0 and p = 7/8. This holds by Eq. 5. Finally, if j = −5/2 the earlier part of the proof shows that we need only check π3/2,−5/2 = 0 when c = 0 and p = 19/16. Since b−1/2,−5/2 is zero here by design, the formula for the singular entries gives π3/2,−5/2 = b3/2,−5/2 β4 (−5/2). We use a trick to avoid evaluating b3/2,−5/2 : Eq. 5 yields π7/2,−5/2 = a7/2,−5/2 α6 (−5/2), and it is known (see [FF80, MP92 or CS04]) that α6 (−5/2) is not a cocycle. Here a7/2,−5/2 = 0, so the cup equation implies that π3/2,−5/2 = 0. √ Part (c). √By the proof of part (b), when c = 0, 2j = −2k − 1 ± 4k 2 + 3, and 2p = ∓ 4k 2 + 3 for some k ∈ 2 + N, we have bj +2,j = bj +2k+2,j +2k = 0. In this p−j
p−j −2k
case both 1,2,3,5,... ,2k+1 and 1,2 are Vec(S 1 )-invariant, and the former contains the latter. For k ∈ 3 + N, their quotient is a jump (4, 2, . . . , 2, 4) extension of length k and parameters (a, j ), and another look at the proof of part (b) shows that it is uniserial. For k = 2 the quotient is a jump 6 extension of length 2. √ To prove that it is uniserial √ one must verify that bj +6,j = 0 for c = 0, 2j = −5 ± 19, and 2p = ∓ 19, a formidable exercise in grade school algebra which we carried out with a computer. Part (d). Here we will not realize the uniserial representations as subquotients, but we will use subquotients to prove their existence. First, use Eq. √ 9 to prove that for c = 0 and k ∈ Z+ , bj +4,j = bj +k+4,j +k = 0 if and only if 4p = ∓ k 2 + 15 and 2j = −k − 3 ± √ √ √ k 2 + 15. We need the case k = 4, so let c = 0, 4p = ∓ 31, and 2j = −7 ± 31. p−j Consider 1,3,5,7 / p−j −9 , a jump (2, 2, 2, 2) extension with parameters (a, j ). By Sect. 5, it is equivalent to a representation on ⊕4r=0 A(a, j + 2r) that acts by a lower triangular 5 × 5 matrix with entries πj +2r,j +2s given by Eq. 2, 0 ≤ r, s ≤ 4. Now it is elementary that β2 (ν +2)∪β2 (ν) and ∂β4 (ν) are zero for any ν ∈ C [Co01]. Hence the cup equation together with bj +8,j +4 = bj +4,j = 0 imply that the 4×4 matrix obtained by deleting the j + 4th row and column above is still a representation, albeit no longer a subquotient. It is easily seen to be uniserial, completing the proof. We remark that if we take k = 2 instead of k = 4, we obtain another subquotient realization of the length 2 jump 6 extension. Part (e). Here we need c = 0, as one of the jumps is odd. First verify that in general, bj +2,j = bj +k,j +k−2 = 0 if (10) −2p = 2j + k − 1 − 6c2 = 2j 2 + 2(k − 1)j + (k − 3), i.e., 2j = −k + 1 ± (k − 2)2 + 3 + 12c2 . A very long calculation (trivial with a computer) shows that in this case, Eq. 8 simplifies to −24j (2j + 3)bj +4,j = (2j + 5)(4j + k − 3)(4j + 5k − 9).
(11)
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Taking k = 7, j = −13/2, p = 7/2, and c2 = −7/4, we obtain bj +2,j = bj +7,j +5 = p−j −5
bj +4,j = 0, so here 1,2,4 / 2 is a jump (3, 4) extension with parameters (a, −13/2). It is uniserial: bj +3,j = 0 by direct computation, and one checks that bj +7,j +3 = 0 in p−j p−j −3 the same way or by the following trick: If bj +7,j +3 were zero, then 1,2 / 1,4 would be a jump (4, 3) extension with parameters (a, −13/2), so its dual would be a jump (3, 4) DO subquotient with parameters (a, 1/2). But j = 1/2 is not a root of Eq. 11 at k = 7. We mention that the other roots of Eq. 11 at k = 7 do not yield anything: at j = −1, p = −2, and c2 = −1, the subquotient is not uniserial because a2,−1 = 0, and at j = −5/2 a b coefficient is involved. p−j
p−j
p−j −4
p−j
p−j −4
is a jump (3, 3) extension whenever πj +2,j = πj +6,j +4 = 0, Part (f). 1,2 / 2 uniserial when πj +3,j = 0 = πj +6,j +3 . Excepting the dual singular cases j = −1 and −4, the result follows from Eqs. 7 and 10. For j = −1, use in addition Eq. 5. is a jump (3, 2, 2) extension whenever πj +2,j and πj +7,j +4 Part (g). 1,2 / 1,3 are zero, uniserial when πj +3,j , πj +5,j +3 , and πj +7,j +5 are non-zero. Equations 6 and 7 show that for c = 0, bj +2,j = bj +k,j +k−3 = 0 if p=
(j + 2)(j − 1) j 3 + (k − 1)j 2 + (k − 3)j + (k − 3) and c2 = j + 3k − 7 j + 3k − 7
(12)
(when k = 0 these conditions reduce to either j = −2, or p = j − 1 and c2 = j 2 ). The reader may check that for generic values of j , and in particular the singular values listed in the theorem, choosing p and γ according to Eq. 12 with k = 7 yields a uniserial representation. For this it is convenient to use Eq. 12 with k = 0 to verify that bj +3,j = 0, Eq. 10 with k = 5, 7, and Lemma 6.2. For the singular cases j = −1, −7/2, and −11/2 use also Eq. 5. Note that we do not obtain an extension at j = −2; here b−2,1 is involved. p−j
p−j −6
is a jump (3, 2, 3) extension whenever bj +2,j , bj +8,j +6 and Part (h). 1,2,4 / 2 bj +4,j are zero, so Eq. 11 with k = 8 proves the result: the root j = −5/4 does not give a uniserial representation but the root j = −31/4 does, and its dual gives the case γ = 3/4. p−j
p−j −5
Parts (i) and (j). 1,2,4 / 1,3 is a jump (3, 3, 2) extension whenever bj +2,j , bj +8,j +5 and bj +4,j are zero. Another very long calculation using Eq. 12 shows that when bj +2,j and bj +k,j +k−3 are zero, Eq. 8 simplifies to bj +4,j =
(k − 3)(j − 1)(2j + 5) 2 8j + (9k − 14)j + 6(k − 4) . 2j (2j + 3)(j + 3k − 7)2
(13)
The resulting roots of bj +4,j at j = 1 and −5/2 are not useful, as b coefficients are √ involved. At k = 8, the roots 8j = −29 ± 649 give the uniserial subquotients claimed in the theorem. To verify that they are √ √ indeed uniserial, use Eq. 12 to obtain 40p = 71 ∓ 3 649 and 160c2 = −139 ± 7 649, and then use Eq. 10, Lemma 6.2, and the following useful fact: k = 0, c = 0, and bj +3,j = bj +k+3,j +k = 0 imply p = 0 and c2 = 1.
(14)
Pseudodifferential Operator Subquotients p−j
655 p−j −6
Part (j) goes similarly. 1,2,4 / 1,3 is a jump (3, 2, 2, 2) extension whenever bj +2,j , bj +9,j +6 and bj +4,j are zero. Repeating√the above analysis proves √ the exis3529, 80p = 169 ∓ 3 3529, and tence of uniserial subquotients at 16j = −67 ± √ 2 640c = −851 ± 17 3529. p
p−4
Part (k). The jump (1, 2, 3) extension with parameters (a, 0) is 2 / 2 at p = −5/2, p p−3 c2 = −1/2. The jump (1, 3, 2) extension with parameters (a, 0) is 2 / 1,3 at p = p
p−4
−2/11, c2 = 3/11. The jump (1, 2, 2, 2) extension with parameters (a, 0) is 2 / 1,3 at p = −1/7, c2 = 2/7. The jump (2, 1, 3) extension with parameters (a, −2) is p+2 p−2 1 / 2 at generic values of p and c on the parabola 3c2 − 5p = 7. The jump p+2 p−2 (2, 1, 2, 2) extension with parameters (a, −2) is 1 / 1,3 at generic values of p and c on the parabola c2 + 3p = 1. The jump (2, 1, 4) extension with parameters (a, −2) p+2 p−2 is 1 / 3 at p = 5, c2 = 16. The jump (3, 1, 2, 2) extension with parameters p+3 p−2 (a, −3) is 1,2 / 1,3 at p = 2/7, c2 = 13/7. In all of these cases, we used Eqs. 10, 12, 14, and Lemma 6.2 to find the given values of p and c2 and prove uniseriality. 7. Remarks We conclude with several remarks. First, consider the subalgebra Vec(R) of Vec(S 1 ) with basis {zn D : n ∈ N}. For any γ ∈ C, the subspace A+ (γ ) of A(γ , γ ) spanned by {dzγ zn : n ∈ N} is invariant under Vec(R), and it is irreducible for γ = 0 (A+ (0) is a uniserial extension of A+ (1) → D0 ). It is not hard to check that all of the results of this paper still hold when Vec(R) and A+ (γ ) replace Vec(S 1 ) and A(a, γ ). (For example, this can be proven using the methods of the final section of [Co01].) This is the setting of [FF80]. The idea of realizing bounded uniserial modules as subquotients of DO modules is already present in [FF80], where it is noted that 2 A+ (γ ) is generically a uniserial extension of A+ (2γ + 1) → A+ (2γ + 3) → A+ (2γ + 5) → · · · . This is essentially part (a) of our main theorem, as the Adler trace, or non-commutative residue (see for example [CS04]) shows that ⊗2 A+ (γ ) is essentially isomorphic to −1 (1 − γ , 2γ − 1). This module is special, and its decomposition as given by Lemma 5.3 corresponds to the decomposition of ⊗2 A+ (γ ) into its symmetric and antisymmetric subspaces. At the time of [FF80], it was not possible to go beyond part (a) of our theorem because the results of [CMZ97] were not yet available. We mentioned in the introduction that [Co01] gives a computationally effective way to prove that a given composition series does not admit a uniserial extension. Let us describe it in the regular case. Suppose that σ is a uniserial extension of A(a, γ1 ) → · · · → A(a, γn ), where γi is neither γj or 1 − γj whenever i = j . It is proven in [Co01] that up to equivalence we must have γi − γi−1 ∈ 2 + N for all i, and the matrix entries σij : A(a, γj ) → A(a, γi ) must be sij βγi −γj (γj ) for some scalars sij . Uniseriality implies that the si,i−1 are non-zero, so up to equivalence they are all 1. Consider evaluating the (i, j )th entry of the cup equation at z−1 D ∧ z2 D. Both sides will be of the form (z−1 dz)γi −γj times a polynomial in zD, and these polynomials can be obtained with a computer. Given a composition series which in fact does not admit uniserial extensions, one can prove that this is the case by checking that there is no way to choose the sij so that all entries of the cup equation are satisfied at z−1 D ∧ z2 D. (It seems to be true that in order for some choice of the sij to give a representation, it is both
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necessary and sufficient for the cup equation to hold on this one element of 2 Vec(S 1 ). This has not yet been proven, but it is partially explained in [Co01]. In any case, is not needed to prove negative results.) Let us now discuss some aspects of the main theorem. √ First, it appears likely that the uniserial jump (2, 4, 2) extensions with 2γ = −7 ± 31 in part (d) are not subquotients of any DO module, but we have not proven this (messy calculations arise in the c = 0 cases). There are uniserial jump (2, 2, . . . , 2) extensions which are not subquotients of DO modules, as since β2 ∪β2 is zero we can construct such extensions by putting β2 in every matrix entry on the first subdiagonal and zeroes along all the lower subdiagonals. However, it can be shown that such extensions are limits of the extensions in part (a) as p goes to ∞. Second, it is somewhat surprising that our method does not yield any uniserial jump (2, 3, 2) extensions, as the analysis at the end of [Co01] leads one to expect their exisp−j p−j −4 tence. There are two types of subquotients where they could arise: first, in 1,3 / 1,3 p−j
p−j −3
when bj +3,j and bj +7,j +4 are both zero, and second, in 1 / 2,4 when bj +5,j +3 , bj +7,j +3 , and bj +7,j +4 are all zero (or in the dual of this case). Uniseriality fails in the first case because Eq. 14 forces bj +5,j +2 = 0. To analyze the second case, use Eq. 13 with k = 4 and j + 3 replacing j . The roots of the key quadratic are j + 3 = 0 and −11/4. The root j = −3 is useless because b coefficients arise, and j = −23/4 fails because here bj +7,j +5 turns out to be zero. Third, we mention that with a little extra work we could probably obtain a few more uniserial subquotients. For example, we expect that most of the uniserial jump 4 and jump (4, 2) extensions excluded in the exceptions to part (b) are subquotients of DO modules with c = 0. In particular, the uniserial extension of A(a, 1) → A(a, 5) can p−1 p−3 probably be realized as 1,2 / 1 for some values of p and c (one needs only b3,1 = 0 and b5,1 = 0, so this would be simple to check). We single out this case because (at a = 0) it is equivalent to the uniserial extension of A(a, 0) → A(a, 5). It may be that the latter extension arises as a subquotient directly: Eq. 10 gives b2,0 = b5,3 = 0 when p p−1 p = −2 and c2 = −1/3, and miraculously b5,1 is also zero here, so 2 / 1,4 is an extension of A(a, 0) → A(a, 5). However, we did not check its uniseriality because this requires evaluation of b5,0 . We conclude by explaining the presence of the factor k − 3 in Eq. 13. Recall the well known fact that βq (ν) is a cocycle for generic ν if and only if q is 2, 3, or 4. Suppose that bj +2,j and bj +3,j are zero. Then the (j + 5, j )th entry of the cup equation implies that generically, bj +5,j = 0. Moreover, the (j + 7, j )th entry of the cup equation implies that bj +7,j +4 bj +4,j is generically zero, as uniserial jump (4, 3) extensions do not exist for generic values of γ [Co01]. If bj +7,j +4 = 0 then Eq. 14 essentially says that c = 0, so for c = 0 we must have bj +4,j = 0. This forces the presence of the factor k − 3. This factor means that whenever bj +2,j and bj +3,j are both zero, bj +4,j is also zero. In this case it follows easily from the cup equation that bj +k,j = 0 for all k ∈ Z+ , and so A(j ) is essentially a submodule of p−j . However, this does not lead to anything surprising: we must have c = 0 to avoid singular entries, so by Eq. 12 it only occurs in −1 (1, p) and −1 (−p, p). Consequently it has a simple explanation based on the fact that the intertwining map dzD : A(0) → A(1) is a differential operator of order 1. Acknowledgements. This paper grew from a question I first heard asked by J´erˆome Germoni during a brief but fruitful visit to Institut Gerard Desargues at the Universit´e de Lyon I. I thank him and the other members of the institute, in particular Valentin Ovsienko and Olivier Mathieu, for productive discussions.
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References [CMZ97] Cohen, P., Manin, Y., Zagier, D.: Automorphic pseudodifferential operators. In: Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl. 26, Boston: Birkh¨auser, 1997, pp. 17–47 [Co01] Conley, C.: Bounded length 3 representations of the Virasoro Lie algebra. Internat. Math. Res. Notices 2001(12), pp. 609–628 [CS04] Conley, C., Sepanski, M.: Singular projective bases and the affine Bol operator. Adv. Appl. Math. 33(1), 158–191 (2004) [FF80] Feigin, B.L., Fuks, D.B.: Homology of the Lie algebra of vector fields on the line. Func. Anal. Appl. 14(3), 201–212 (1980) [Ga00] Gargoubi, H.: Sur la g´eom´etrie de l’espace des op´erateurs diff´erentiels lin´eaires sur R. Bull. Soc. Roy. Sci. Li`ege 69(1), 21–47 (2000) [Ge01] Germoni, J.: On the classification of admissible representations of the Virasoro Lie algebra. Lett. Math. Phys. 55(2), 169–177 (2001) [LO99] Lecomte, P., Ovsienko, V.: Projectively invariant symbol calculus. Lett. Math. Phys. 49(3), 173–196 (1999) [Ma92] Mathieu, O.: Classification of Harish-Chandra modules over the Virasoro Lie algebra. Invent. Math. 107(2), 225–234 (1992) [MP91] Martin, C., Piard, A.: Indecomposable modules over the Virasoro Lie algebra and a conjecture of V. Kac. Commun. Math. Phys. 137, 109–132 (1991) [MP92] Martin, C., Piard, A.: Classification of the indecomposable bounded modules over the Virasoro Lie algebra with weightspaces of dimension not exceeding two. Commun. Math. Phys. 150, 465–493 (1992) Communicated by A. Connes
Commun. Math. Phys. 257, 659–665 (2005) Digital Object Identifier (DOI) 10.1007/s00220-004-1266-5
Communications in
Mathematical Physics
Factoriality of q-Gaussian von Neumann Algebras ´ Eric Ricard D´epartement de Math´ematiques de Besan¸con, Universit´e de Franche-Comt´e, 25030 Besan¸con cedex, France. E-mail: [email protected] Received: 28 May 2004 / Accepted: 8 July 2004 Published online: 13 January 2005 – © Springer-Verlag 2005
Abstract: We prove that the von Neumann algebras generated by n q-Gaussian elements, are factors for n 2.
1. Introduction In the early 70’s, Frish and Bourret considered operators satisfying the q-canonical commutation relations, for −1 < q < 1 : l(e)l ∗ (f ) − ql ∗ (f )l(e) = (e, f )I d. Nevertheless their existence was proved only 20 years later by Bo˙zejko and Speicher in [2]. Since then, many people studied the von Neumann algebra q (HR ), generated by q-Gaussian random variables {l(e) + l ∗ (e); e ∈ HR }, and some of their generalizations. It is well known that q (HR ) is type I I1 . One of the interesting points is that these algebras realize a kind of interpolating scale between 1 (H) which is commutative, and −1 (H ), the hyperfinite I I1 factor. For q = 0, we recover the algebra generated by Voiculescu’s semicircular elements, which is a central object in the free probability theory. Among the known results, Bo˙zejko and Speicher showed that q (HR ) is non-injective under some condition on the dimension of H, which was removed by Nou [4]. Recently, Shlyakhtenko [5] proved that they are solid for some values of q. The question of the factoriality of q (HR ) was studied by Bo˙zejko, K¨ummerer and Speicher [1]; they showed that if H is infinite dimensional then q (HR ) is a factor. This condition was partially ´ released by Sniady [6], who showed that this is still true if the dimension of H is greater than a function of q.
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2. Preliminaries In this paper, −1 < q < 1 is a fixed real number, we will use standard notation and refer to the papers [3, 1, 4] for general background. Let H be the complexification of some real Hilbert space HR . By H⊗2 n (n 1), we denote the hilbertian n-tensor product of H with itself; this space is equipped with a scalar product that we write (., .). Let Pn : H⊗2 n → H⊗2 n be given by Pn (e1 ⊗ . . . ⊗ en ) = q |σ | eσ (1) ⊗ . . . ⊗ eσ (n) = q |σ | φ(σ )(e1 ⊗ . . . ⊗ en ), σ ∈Sn
σ ∈Sn
where Sn is the symmetric group on n elements, |σ | is the number of inversion of σ , and φ is the natural action of Sn on H⊗2 n . It was shown in [3], that this operator is bounded and strictly positive; therefore we denote by H⊗n , the Hilbert space H⊗2 n equipped with the new scalar product ., . given by ∀x, y ∈ H⊗n
x, y = (x, Pn (y)).
From now on, if x ∈ H⊗n , x is the norm of x with respect to this new scalar product. For instance, if e ∈ H and e = 1, then e⊗n 2 = [n]q !, where [k]q =
1−q k 1−q
and [n]q ! = [1]q . . . [n]q .
Remark 1. We will use as a key point that the sequence ([n]q !) behaves like a geometric sequence. Moreover, it is known that the following algebraic relation holds : Pn = Rn,k (Pn−k ⊗ Pk ) with Rn,k = q |σ | φ(σ −1 ), σ ∈Sn /Sn−k ×Sk
and the sum runs over the representatives of the right cosets of Sn−k × Sk in Sn with a minimal number of inversions. As a consequence, since Rn,k B(H⊗2 n ) Cq = i −1 i 1 (1 − |q| ) , we get that the formal identity map
has norm bounded by
I d : H⊗n−k ⊗2 H⊗k → H⊗n Cq .
Remark 2. As an application, we get that, if e1 , . . . en and e are norm 1 vectors in H, then n/2 e1 ⊗ . . . ⊗ en ⊗ e⊗m H⊗n+m Cq [m]q !. The q-deformed Fock space is the Hilbert space defined by Fq (HR ) = C ⊕ ⊕n1 H⊗n , where is a unital vector, considered as the vacuum. Vectors in H will be called letters and an elementary tensor of letters in H⊗n will be called a word of length n.
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For e ∈ HR , we consider left and right creation operators on Fq (HR ), given by : l(e)(e1 ⊗ . . . ⊗ en ) = e ⊗ e1 ⊗ . . . ⊗ en , lr (e)(e1 ⊗ . . . ⊗ en ) = e1 ⊗ . . . ⊗ en ⊗ e. They are bounded endomorphisms of Fq (HR ); more precisely if e = 1 then 1 if q 0 lr (e) = l(e) = √ 1 if q 0 . 1−q
Their adjoints in B(Fq (HR )) are the annihilation operators : l ∗ (e)(e1 ⊗ . . . ⊗ en ) = q i−1 (e, ei ) ⊗ e1 ⊗ . . . ⊗ eˆi ⊗ .. ⊗ en , 1i n
lr∗ (e)(e1
⊗ . . . ⊗ en ) =
q n−i (e, ei ) ⊗ e1 ⊗ . . . ⊗ eˆi ⊗ .. ⊗ en ,
1i n
where eˆi denotes a removed letter; if n = 0, we put l ∗ (e) = lr∗ (e) = 0. The operators l(e) satisfy the q-commutation relations : l(e)l ∗ (f ) − ql ∗ (f )l(e) = (e, f )I d. For e ∈ HR , let W (e) = l(e) + l ∗ (e)
and
Wr (e) = lr (e) + lr∗ (e).
So for e ∈ HR , W (e) is self-adjoint. q (HR ) stands for the von Neumann algebra generated by (W (e))e∈Hr , q (HR ) = { W (e) ; e ∈ HR } . And, q,r (HR ) stands for the von Neumann algebra generated by (Wr (e))e∈HR q,r (HR ) = { Wr (e) ; e ∈ HR } . We recall some classical results on those algebras, – The commutant of q (HR ) is q (HR ) = q,r (HR ). – The vacuum vector is separating and cyclic for both q (HR ) and q,r (HR ). – The vector state τ (x) = x, is a trace for both q (HR ) and q,r (HR ). According to the second point, any x ∈ q (HR ) is uniquely determined by ξ = x. ∈ Fq (HR ), so we will call it x = W (ξ ) (and similarly for q,r (HR ), x = Wr (ξ )). This notation is consistent with the definition of W (e) = l(e) + l ∗ (e). The subspace q (HR ). ⊂ Fq (HR ) of all such ξ contains all tensors of finite rank, so it contains all words. If e1 ⊗. . .⊗en is a word in Fq (HR ), there is a nice description of W (e1 ⊗. . .⊗en ) in terms of l(ei ) called the Wick formula : W (e1 ⊗ . . . en ) =
n
q |σ | l(eσ (1) ) . . . l(eσ (n−m) )
m=0 σ ∈Sn /Sn−m ×Sm ×l ∗ (eσ (n−m+1) ) . . . l ∗ (eσ (n) ),
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where σ is the representative of the right coset of Sn−m × Sm in Sn with a minimal number of inversions. There is a similar formula for Wr . Actually, the algebras q (HR ) and q,r (HR ) are in standard form in B(Fq (HR )), but we won’t use it. If we denote by S the anti-symmetry that inverses the order of words in HR , then for any ξ ∈ q (HR ). : W (ξ )∗ = W (Sξ )
and
S.W (ξ ).S = Wr (Sξ ).
In particular q (HR ). = q,r (HR ).. Remark 3. For ξ, η ∈ q (HR )., we will frequently use W (ξ )η = W (ξ )Wr (η) = Wr (η)W (ξ ) = Wr (η)ξ. Let T : HR → HR , be a R-linear contraction, then there is a canonical C-linear contraction, Fq (T ), on Fq (HR ) extending T , called the first quantization; formally Fq (T ) = I dC ⊕ ⊕n1 T˜ ⊗n with T˜ , the complexification of T on H. The second quantization of T , is the unique unital completely positive map q (T ) on q (HR ) satisfying, for ξ ∈ q (HR ). q (T )(W (ξ )) = W (Fq (T )ξ ). For instance, if KR ⊂ HR , the second quantization associated to the orthogonal projection PKR on KR is a conditional expectation q (PKR ) : q (HR ) → q (KR ) = {W (e) ; e ∈ KR } . 3. The Main Result Let e ∈ HR of norm one and denote by Ee the closed subspace of Fq (HR ) spanned by the elements {e⊗n ; n 0}, that is Ee = Fq (Re). It is easy to check that for any x = W (ξ ) ∈ W (e) , we have ξ ∈ Ee . Conversely, assume x = W (ξ ) and that ξ ∈ Ee ; then x ∈ W (e) : by the second quantization, we have a conditional expectation q (PRe ) : q (HR ) → W (e) , but then q (PRe )(x). = Fq (PRe ).ξ = PEe .ξ = ξ = x., as is separating, x = q (PRe )(x) ∈ W (e) . Theorem 1. Assume that dim H 2 and let e ∈ HR , e = 1, then W (e) is a maximal abelian sub algebra in q (HR ). Corollary 1. q (HR ) is a factor as soon as dim H 2. Proof. Let x ∈ q (HR ) ∩ q (HR ) , then there is ξ ∈ Fq (HR ) such that x = W (ξ ). By the theorem, we must have x ∈ W (e) for every e ∈ HR , but then ξ ∈ Ee , so necessarily x ∈ C.
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Proof. Fix (ei )i 0 an orthonormal basis in HR , with e0 = e. Let x = W (ξ ) ∈ q (HR ) ∩ W (e) , we have to show that ξ ∈ Ee . For any y = W (η) with η ∈ Ee , we have xy − yx = 0, (W (ξ )W (η) − W (η)W (ξ )). = 0, (Wr (η) − W (η))ξ = 0. So ξ ∈ ∩y=W (η)∈W (e) ker (Wr (η) − W (η)). By duality, we have to prove that span{ran (Wr (η) − W (η)) ; y = W (η) ∈ W (e) } ⊃ Ee⊥ . Ee⊥ is the closed linear span of the set of elementary tensors F = {ei1 ⊗ . . . ⊗ ein ; n 1, and (i1 , . . . , in ) ∈ Nn \{(0, . . . , 0)}}. Let z = ei1 ⊗ . . . ⊗ ein be a word in F ; it suffices to prove that z is a weak-limit of elements in span{ran (Wr (η) − W (η)) ; y = W (η) ∈ W (e) }. The von Neumann algebra W (e) is commutative and diffuse and separably generated (see [1]), so we can assume that W (e) = L∞ ([0, 1], dm), where dm is the Lebesgue measure. With this identification, the Rademacher functions ri belong to W (e) , so we have ri = W (ηi ) for some ηi ∈ Ee . Obviously W (ηi ) is a self-adjoint symmetry and W (ηi )2 = 1. Moreover, the sequence (ηi )i 1 converges to 0 for the weak topology on Fq (HR ), since ri is an orthonormal system in L2 ([0, 1], dm). Consider zi = (W (ηi ) − Wr (ηi ))(W (ηi )(z)), obviously zi ∈ span{ran (Wr (η) − W (η)) ; y = W (η) ∈ W (e) }, and a simple calculation gives zi = W (ηi )2 (z) − Wr (ηi )W (ηi )(z) = z − Wr (ηi )W (ηi )(z). We will show that yi = Wr (ηi ).W (ηi )(z) tends weakly to 0 in Fq (HR ). As yi z, it suffices to prove that for any word t = ej1 ⊗ . . . ⊗ ejp , yi , t → 0. We have, yi , t = Wr (ηi ).W (ηi )(z), t = Wr (z)(ηi ), W (t)(ηi ). This is the point where we use the Wick formula : n W (ej1 ⊗ . . . ⊗ ejn ) = q |σ | l(ejσ (1) ) . . . l(ejσ (n−m) ) m=0 σ ∈Sn /Sn−m ×Sm ×l ∗ (ejσ (n−m+1) ) . . . l ∗ (ejσ (n) ),
and similarly for z. Since the number of terms appearing after developing the sums is finite (it depends only on n and p), we only need to show that Ii = lr (ei1 ) . . . lr (eim )lr∗ (eim+1 ) . . . lr∗ (ein )(ηi ), l(ej1 ) . . . l(ejr ) ×l ∗ (ejr+1 ) . . . l ∗ (ejp )(ηi ) → 0, as soon as at least one of the ik ’s is non-zero. Let v be the first k such that ik = 0. Since the letters in ηi are only e, we can suppose that v m, otherwise lr (ei1 ) . . . lr (eim )lr∗ (eim+1 )
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. . . lr∗ (ein )(ηi ) = 0 (we have to cancel some eiv in ηi !). More generally, we can assume that eim+1 = . . . = ein = ejr+1 = . . . = ejp = e. Recall that l(e)∗ e⊗n = [n]q e⊗n−1 . Now, we write ηi = k 0 aki e⊗k , interchanging the sums and making simplifications gives that (with a−n = 0 if n > 0. The an are reals since ri is self adjoint), Ii = lr (ei1 ) . . . lr (eim )lr∗ (eim+1 ) . . . lr∗ (ein )(ηi ), l(ej1 ) . . . l(ejr )l ∗ (ejr+1 ) . . . l ∗ (ejp )(ηi ) =
i i ak+n−2m ak+p−2r [k + n − 2m]q !/[k − m]q ![k + p − 2r]q !/[k − r]q !
k r,m
.lr (ei1 ) . . . lr (eim )e⊗k−m , l(ej1 ) . . . l(ejr )e⊗k−r =
i i ak+n−2m ak+p−2r [k + n − 2m]q !/[k − m]q ![k + p − 2r]q !/[k − r]q !
k r,m
.lr (eiv+1 ) . . . lr (eim )e⊗k−m , lr∗ (eiv ) . . . lr∗ (ei1 )(ej1 ⊗ . . . ⊗ ejr ⊗ e⊗k−r ). Assume that k is big (say k > N > 2(n + p)); by the definition of v, we have that i1 = . . . = iv−1 = e, so lr∗ (eiv−1 ) . . . lr∗ (ei1 )(ej1 ⊗ . . . ⊗ ejq ⊗ e⊗k−r ) is obtained by canceling (v − 1) times the letter e in the word ej1 ⊗ . . . ⊗ ejr ⊗ e⊗k−r using some geometric weight q α , ... δh1 ,... q ( hi )−v+1 (ej1 ⊗. . .⊗ejr ⊗e⊗k−r )(h1 ,... ,hv−1 ) , 1hv−1 k−v−2
1h2 k−1 1h1 k
where (ej1 ⊗ . . . ⊗ ejr ⊗ e⊗k−r )(h1 ,... ,hv−1 ) is obtained from ej1 ⊗ . . . ⊗ ejr ⊗ e⊗k−r by removing the letter on the h1 th position from the right, then the letter at the h2 th position in the remaining word and so on, and where δh1 ,... is one if all the removed letters are e and 0 otherwise. To have a non-zero term in lr∗ (eiv )(ej1 ⊗ . . . ⊗ ejr ⊗ e⊗k−r )(h1 ,... ,hv−1 ) we have to cancel a letter that is not an e, so it can happen only for the terms coming from ej1 ⊗ . . . ⊗ ejr (if there are some left !); as this word of length k − v + 1 ends with at least (k − r − v + 1) e, we end up with a sum of at most r words in front of which there is a factor less than |q|k−r−v+1 . Moreover, by Remark 2, the norm of such a word r/2 is less than Cq [k − r − v + 1]q !. If we sum up everything, we get that lr∗ (eiv ) . . . lr∗ (ei1 )(ej1 ⊗ . . . ⊗ ejq ⊗ e⊗k−r ) C(n, m, v, q)|q|k [k]q !, where C(n, m, v) does not depend on k (because [k]q Cq ). Now we can estimate Ii , by cutting the sum into two parts Ai + Bi = k N |.| + k N |.|.
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Since ηi → 0 weakly, each aji tends to 0, then Ai
i→∞
i i |ak+n−2m ||ak+p−2r |C(k, n, p) → 0,
N>k r,m
and as ηi 1, we have |aki | 1/ [k]q !, so [k + n − 2m]q ![k + p − 2r]q ! lr∗ (eiv ) . . . lr∗ (ei1 ) Bi [k − m]q ! [k − r]q ! k N
×(ej1 ⊗ . . . ⊗ ejq ⊗ e⊗k−q ).lr (eiv+1 ) . . . lr (eim )e⊗k−m [k + n − 2m]q ![k + p − 2r]q ! C|q|k [k]q !C(q)m [k − m]q ! [k − m]q ! [k − r]q ! k N C|q|k C|q|N . k N
Consequently, we get that lim sup |Ii | C|q|N for every N , so Ii → 0.
References 1. Bo˙zejko, M., K¨ummerer, B., Speicher, R.: q-Gaussian processes: non-commutative and classical aspects. Commun. Math. Phys. 185(1), 129–154 (1997) 2. Bo˙zejko, M., Speicher, R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137(3), 519–531 (1991) 3. Bo˙zejko, M., Speicher, R.: Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann. 300(1), 97–120 (1994) 4. Nou, A.: Non injectivity of the q-deformed von Neumann algebras. Math. Ann. 300(1), 17–38 (2004) 5. Shlyakhtenko, D.: Some estimates for non-microstates free entropy dimension, with applications to q-semicircular families. http://arxiv.org/abs/math.OA/0308093, 2003 ´ 6. Sniady, P.: Factoriality of Bo˙zejko–Speicher von Neumann algebras. Commun. Math. Phys. 246(3), 561–567 (2004) Communicated by Y. Kawahigashi
Commun. Math. Phys. 257, 667–701 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1356-z
Communications in
Mathematical Physics
On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation Lixin Tian1 , Guilong Gui1 , Yue Liu2 1
Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R. China 2 Department of Mathematics, University of Texas, Arlington, TX 76019, USA Received: 29 May 2004 / Accepted: 2 December 2004 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as α 2 → 0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation. 1. Introduction In [1], Camassa and Holm used Hamiltonian methods from physical principles (also see [2, 3]) to derive the nonlinear dispersive wave equation ut + 2ωux − uxxt + 3uux = 2ux uxx + uuxxx
(1.1)
by retaining two terms that are usually neglected in the small amplitude shallow water limit (which gives the Korteweg de Vries), where u is the fluid velocity in the x direction (or equivalently the height of the free surface of water above a float bottom), ω is a constant related to the critical shallow water wave speed. They show that for all ω, (1.1) has a Lax pair formulation, and for ω=0, (1.1) has travelling wave solutions of the form ce−|x−ct| , which is called peakon because they have a discontinuous first derivative at the wave peak. For every ω, Eq. (1.1) is bi-Hamiltonian and thus possesses an infinite number of conservation laws. Moreover, Eq. (1.1) has a simple multi-peakon which shows many fantastic properties. Dullin, Gottwald, Holm[4] discussed the following 1+1 quadratically nonlinear equation in this class for a unidirectional water wave with fluid velocity u(x, t), mt + c0 ux + u mx + 2 m ux = −γ uxxx ,
x ∈ R,
t ∈ R,
(1.2)
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L. Tian, G. Gui, Y. Liu
where m = u − α 2 uxx is√a momentum variable, the constants α 2 and γ /c0 are squares of length scales, and c0 = gh (where c0 := 2ω) is the linear wave speed for undisturbed water at rest at spatial infinity. Equation (1.2) was derived by using asymptotic expansions directly in the Hamiltonian for Euler’s equations in the shallow water regime and thereby shown to be bi-Hamiltonian and has a Lax pair formulation in [1, 4]. Equation (1.2) combines the linear dispersion of the Korteweg-de Vries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and CH equation as limiting cases. In [4], there are two main results reported. Firstly, they identified how the dispersion coefficients for the linearized water waves appear as parameters in the isospectral problem for these IST-integrable shallow water wave equations, demonstrated that its γ phase speed lies in the band − α 2 , 2 ω and longer linear waves are the faster provided 2α 2 ω + γ ≥ 0, and determined how the linear dispersion parameters α,c0 and γ in (1.2) affect the isospectral content of its soliton solutions and the shape of its travelling waves. Secondly, Eq. (1.2) was rederived by using a certain nonlocal form of shallow water wave asymptotic that is correct to one order higher than for KdV. This new derivation and analysis in [4] attached additional physical meaning to Eq. (1.2) in the context of asymptotic for shallow water wave equations. Hence, Eq. (1.2) is a new and important integrable shallow water wave equation. It has a soliton-fliking KdV equation and peakon-liking Camassa-Holm equation. In this paper, we are interested in the study of some properties of the solutions of the Cauchy problem and the scattering problem associated to (1.2). Now we denote (1.2) for the DGH equation. Using the notation m = u − α 2 uxx , one can rewrite the initial value problem of the DGH equation as ut − α 2 uxxt + 2ω ux + 3uux + γ uxxx = α 2 (2ux uxx + uuxxx ) , t > 0, x ∈ R, u (0, x) = u0 (x) . (1.3) If the weak dispersive term γ uxxx is rewritten as the strong dispersive term γ u − α 2 uxx xxx , then we get ut − α 2 uxxt + 2ω ux + 3uux + γ u − α 2 uxx xxx = α 2 (2ux uxx + uuxxx ) , t > 0, x ∈ R, u (0, x) = u0 (x) . (1.4) Equation (1.3) is connected with two separately integrable soliton equations for water waves. When α 2 → 0, this equation becomes the Korteweg-de Vries (KdV) equation ut + 2ωux + 3uux = −γ uxxx . While ω = 0, it is well known that there exists a smooth soliton. Instead, taking γ → 0 in Eq. (1.3), it turns out to be the Camassa-Holm equation ut + 2ωux − α 2 uxxt + 3uux = α 2 (2ux uxx + uuxxx ).
(1.5)
Many researches have been carried out on the Camassa-Holm equation [1-11]. In [5, 6], numerical simulations and the conserved quantities of (1.5) are investigated. In
Well-Posedness Problem and Scattering Problem for the DGH Equation
669
[7], symmetries and the integrable perturbation of (1.5) are discussed. In [9], the soliton solution of (1.5) is investigated by using a variation method. Tian, et al. in [10] discussed the traveling wave solutions and double soliton solutions of (1.5), and introduced the definitions of concave, convex peaked soliton and smooth soliton solutions. Tian, Song, Yin [11, 12] considered the generalized Camassa -Holm equation and derived some new exact peakon and compacton. In [13–15], Constantin and Escher studied the global existence, blow-up of the solution and Hamiltonian structure for Camassa-Holm equation. And in [16], A. Constantin and H.P. Mckean pointed out that the isospectral problem for the limiting case ω = 0 of (1.1) is completely different from the one when ω = 0. In [17], Constantin and Strauss proved that the solitary waves possess the spectral properties of solitons of Camassa-Holm and that their shapes are stable under small disturbances. In [17–19], A. Constantin, J. Lenells, R. Beals, D. Sattinger, and J. Szmigielski studied the scattering problem for the Camassa-Holm equation. They gave an exact description of the spectrum of Eq. (1.1) in L2 (R) under the assumption that the initial momentum m0 ∈ H 1 (R) satisfies (1 + |x|) |m0 (x)| dx < ∞.They also R
got the determination of the evolution under the Camassa-Holm flow of the scattering data associated to an initial profile for (1.1) in the absence of bound states, and proved that Eq. (1.1) is integrable for a given class of initial potentials. In [19], Lenells gave a new scattering approach for the Camassa-Holm Eq. (1.1). Using the method in [18], the scattering approach is implemented to solve the inverse scattering problem for Eq. (1.1). And the approach is best exemplified by the fact that it shows that the solitary waves of (1.1) are solitons. And in [21, 22], A. Constantin, J. Escher and R. Danchin studied wave breaking for nonlinear nonlocal shallow water equations, where wave breaking holds if the solution (representing the wave) remains bounded but its slope becomes infinite in finite time. In [23], Guo and Liu investigated the peaked wave solutions of the DGH equation by using the qualitative analysis methods of planar autonomous systems and numerical simulation, and some explicit expressions of peaked solitary wave solutions and peaked periodic wave solutions are obtained. In [24], Tang and Yang obtained the general explicit expressions of the two wave solutions of Eq. (1.2) by using the bifurcation phase portraits of the traveling wave system. And in [25], J. Bona and R. Smith demonstrated the KdV equation as the limit of the BBM equation. On the basis of the above researches, this paper studies the Cauchy problem and scattering problem for the DGH equation (1.2) by studying the isospectral problem associated to (1.2), the solution of the initial-value problem, the behavior of solutions of the Cauchy problem for (1.3) as the dispersive parameter γ tends to zero, and the exact peaked solitary wave solutions. We can prove that Eq. (1.3) has a global solution and the solitary waves of Eq. (1.3) are stable. Given some priori estimates, the locally strong limit of the solution of the DGH equation as the dispersive parameter γ tends to zero is obtained. By using the method in [36], we can demonstrate that the solutions of the DGH equation as α 2 tends to zero convergence to the solution of the corresponding KdV equation. And the scattering data of the scattering problem for Eq. (1.2) is expressed. Finally, the new exact peaked solitary wave solutions of the DGH equation are obtained by using a direct method. We give the condition for existing peakon in the DGH equation, which turns out to be nonlinearly stable for the peakon in the DGH equation. Notations. We shall use the standard notation |•|p for the norm of the space Lp ,1 ≤ p ≤ 1/p . The space L∞ = L∞ (R) consists of all essentially ∞, i.e., |f |p = R |f |p dx bounded, Lebesgue measurable functions f with the standard norm |f |∞ = |f |L∞ =
670
inf
L. Tian, G. Gui, Y. Liu
sup |f (x)|. And we denote the norm in the Sobolev space H s by
m(e)=0 x∈R\e
f s = f H s =
R
1/2 s ˆ 2 1 + |ξ | f (ξ ) dξ 2
for s ∈ R. Here fˆ (ξ ) is the Fourier transform of f . We also define the operator s for s any integer s by the formula s = 1 − ∂x2 2 and denote •, • s as the inner product on H s. The reminder of the paper is organized as follows: In Sect. 2, we study the global well-posedness of the Cauchy problem (1.3). The local well-posedness for (1.3) in H s , s > 23 can be obtained easily by Kato’s theory [26] for quasilinear evolution equations. Under the assumption of a simple condition on the initial data, however , the higher Sobolev space H s , s ≥ 3 is required to obtain the global existence for (1.3), since there exist certain classes of initial data posed on H s , s > 23 , which lead to the solutions of (1.3) that form singularities in finite time [13]. The proof of the global existence is based in the fiction of Constantin and Escner’s proof provided the initial data u0 satisfies certain positively conditions, especially, m0 + ω + 2αγ 2 ≥ 0. The key ingredient used here is to show the uniformly boundedness of |∂x u|∞ . In Sect. 3, the issue of passing to the limit as γ → 0 is investigated. Using the global estimate of the uniform bound for |ux |, it is shown that the solution u of (1.3) with respect to γ is a Cauchy sequence in L2 and therefore is convergent to the solution of the Camassa-Holm equation in H s ,s ≥ 3. The convergence of solutions to the DGH equation as α 2 → 0 is studied in Sect. 4, which contributed to the method in [36]. Section 5 is devoted to proving the stability of solitary waves in H 1 for the DGH equation and get that the solitary waves tend to peakon in condition 2ωα 2 + γ → 0 for DGH equation. To establish the result in view, we follow the general approach to stability pioneered by Grillakis, Shatah and Strauss [27], but using the detailed analysis of solitary waves by Constantin and Strauss [17]. In Sect. 6, the scattering a data of the scattering problem for Eq. (1.3) are expressed by using the standard Gelfand-Dorfman theory[28] and bi-Hamiltonian property of (1.3). Moreover in Sect. 7, we directly obtain a type exact peaked solitary wave solution of the DGH equation. The peakon solution only depends on γ , α, ω. Remark that as ω = 0, the Camassa-Holm equation has peakon solution and as ω = 0, the Camassa-Holm equation has not peakon. But as ω = 0, we obtain that the DGH equation has peakon solution. We give the condition for existing peakon in the DGH equation in Sect. 7. 2. The Initial-Value Problem −1 1 e−| αx | We define p (x) = 2α , x ∈ R, then 1 − α 2 ∂x2 f = p ∗ f for allf ∈ L2 (R), where ∗ denotes convolution, so p ∗ m = u. Using this identity, we can rewrite Eq. (1.3) as the following nonlocal form
α2 2 ut + uux + ∂x p ∗ (u + ω)2 + ux + γ uxx = 0. 2 From [4], Eq. (1.3) is bi-Hamiltonian and, isospectral. The term bi-Hamiltonian means the equation may be written in two compatible Hamiltonian ways, namely, mt = −B2
δE δF = −B1 δm δm
(2.1)
Well-Posedness Problem and Scattering Problem for the DGH Equation
with
671
1 E(u) = u2 + α 2 u2x dx, B2 = ∂x (m + ω) + (m + ω)∂x + γ ∂x3 , 2 1 F (u) = u3 + α 2 uu2x + 2ωu2 − γ u2x dx, B1 = ∂x − α 2 ∂x3 , 2
(2.2)
where B1 , B2 both are Hamilton operators of the DGH equation, and E (u) , F (u) are two conservative laws. Hence Eq. (1.3) has a bi-Hamiltonian structure. Equation (1.3) is suitable for applying Kato’s theory (see [26]), as an outcome, we have Proposition 2.1. Given u0 ∈ H s (R) , s > 23 , there exists a maximal time T (α, ω, γ , u0 ) > 0, and a unique solution u to Eq. (1.3), such that u = u (·, u0 ) ∈ C [0, T ) ; H s (R) ∩ C 1 [0, T ) ; H s−1 (R) ; Moreover, the solution u of (1.3) depends continuously on the initial data u0 . Proposition 2.1 contains as particular cases (for various choices of the constants γ and ω) the well-posedness results obtained in [13, 29, 30]and [31]. According to the inequalities u2 + u2x dx ≤ u2 + α 2 u2x dx = 2E (u) , (α ≥ 1) ,
1 α2
u2 + u2x dx ≤
u2 + α 2 u2x dx =
2 E (u) , α2
(α ≤ 1) ,
the invariance of E (u) ensures that all solutions of Eq. (1.3) are uniformly bounded as long as they exist. Considering the above, we can obtain the following results. Theorem 2.2. Given u0 ∈ H s (R) , s > 23 , the solution u = u(· , u0 ) of Eq. (1.3) is uniformly bounded on [0, T ). Moreover, T < ∞ if and only if lim inf { inf [ux (t, x)]} = −∞, t↑T
x∈R
i.e., singularities can arise only in the form of wave breaking. Proof. The boundedness of the solution is obtained immediately by the invariant E (u). As for wave breaking, let us first assume that u0 ∈ H s (R) for some s ∈ N, s ≥ 4. Multiplying (1.2) with m and integrating on R with respect to x, we obtain d 2 2 m dx = −3 m ux dx − 4ω mux dx − 2γ muxxx dx (2.3) dt R R R R
γ m2 ux dx − 4 ω + 2 mux dx 2α R R
= −3 = −5 R
m2x ux dx − 4
mmx uxx dx − 4ω R
(2.4)
mx uxx dx − 2γ R
mx uxxxx dx. R
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On the other hand, differentiating (1.2) with respect to x and multiplying with mx , integrating on R with respect to x, and integrating by parts yield. It is thereby inferred from (2.3) and (2.4) that d m2 + α 2 m2x dx = − m2 ux dx − 5α 2 m2x ux dx. (2.5) dt R R R If the constant K0 < 0 is the bound on ux from below on [0, T ), we can get that
m2 + α 2 m2x dx , (2.6) − m2 ux dx ≤ −K0 R
R
−5α
2
m2x ux dx
≤ −5K0
2
m2x dx
≤ −5K0
m +α 2
2
m2x dx
.
(2.7)
And by means of Gronwall’s inequality, we can get that
m2 + α 2 m2x dx ≤ m20 + α 2 m20x dx e−6K0 t , f or all t ∈ [0, T ).
(2.8)
R
α R
R
Combining (2.5) with (2.6) and (2.7), we have d 2 2 2 m + α mx dx ≤ −6K0 m2 + α 2 m2x dx. dt R R
R
R
Noting that
u (t)H 3 ≤ max 3α 4 ,
3 α4
1/2 R
m2 + α 2 m2x dx
and according to (2.8), we get that if {ux (t)} is bounded from below on [0, T ), then the H 3 (R)-norm of the solution to (1.3) is said not to have broken in finite time. By Proposition 2.1, we obtain the statement of the proposition for s ∈ N, s ≥ 4. The continuous dependence on initial data ensures the validity for all s > 23 . This completes the proof of Theorem 2.2. We discuss now the question of finite time blow-up of solution to Eq. (1.3) with rather general initial data. For convenience, we assume α = 1 in the following discussion. Let T >0 be the maximal existence time of the solution u (t, •) of Eq. (1.3) with initial data u0 ∈ H s (R) , s ≥ 3, so according to the formulation of Eq. (1.3),
−1
1 2 2 2 ut + uux + ∂x 1 − ∂x (u + ω) + ux + γ uxx = 0, 2 we obtain that
−1
1 γ u2 + u2x + 2 ω + u = 0. ut + uux − γ ux + ∂x 1 − ∂x2 2 2
(2.9)
Differentiating Eq. (2.9) with respect to x, we have
−1
1 γ u = 0. u2 + u2x + 2 ω + utx + uuxx + u2x − γ uxx − ∂x2 1 − ∂x2 2 2
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673
Since ∂x2 p ∗ f = p ∗ f − f , one can see that
1 γ utx = −uuxx − u2x + γ uxx + u2 + u2x + 2 ω + u 2 2
−1 1 γ u u2 + u2x + 2 ω + − 1 − ∂x2 2 2 1 γ = −uuxx + γ uxx + u2 − u2x + 2 ω + u 2 2
−1 1 γ − 1 − ∂x2 u2 + u2x + 2 ω + u (2.10) 2 2 in C [0, T ); L2 (R) . Define now M (t) := inf x∈R [ux (t, x)] and let ξ (t) ∈ R be a point where this infimum is attained. We can easily obtain uxx (t, ξ (t)) = 0, by the definition of ξ (t), since u (t) ∈ H 3 (R) ⊂ C 2 (R). Hence, setting x = ξ (t) in (2.10), we obtain from Theorem 2.1 in [21] the relation dM γ 1 + M 2 = u2 (t, ξ (t)) + 2 ω + u (t, ξ (t)) dt 2 2
−1 1 2 2 2 − 1 − ∂x u (t, ξ (t)) + ux (t, ξ (t)) 2 −1 γ 1 − ∂x2 u (t, ξ (t)) a.e. on (0, T ) . (2.11) +2 ω + 2 Observe that the following inequalities
−1
1 1 2 1 2 2 1 − ∂x u + ux ≥ u2 , u2 < E (u0 ) . 2 2 2 in the proof of Theorem 4.2 in [21, 22] and −1 −1 −1 2 2 2 2 2 1 − ∂x u = u + ∂x 1 − ∂x u ≤ ∂x 1 − ∂x u ∞ −1 2 2 + |u|∞ ≤ u ∂x 1 − ∂x + u1 1
≤ ux 0 + u1 ≤ 2 u1 = 2 u0 1 = 2 [E (u0 )]1/2 . So dM 1 1 ≤ − M 2 + E (u0 ) + |6ω + 3γ | [E (u0 )]1/2 . dt 2 4 Involved manipulations of Eqs. (2.10) and (2.11) and estimates analogous to those in [21, 22] for the special case of the Camassa-Holm equation lead us to the following result. Theorem 2.3. Given u0 ∈ H s (R) , s > 23 , and assume that we can find x0 ∈ R with 1/2 √ γ 1 . u0 (x0 ) < − E (u0 ) + 6 2 ω + [E (u0 )]1/2 2 2 Then wave breaking for the corresponding solution to Eq. (1.3) occurs.
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Now we discuss the existence of global solutions to Eq. (1.3). We get the following theorem. Theorem 2.4. Given u0 ∈ H s (R) , s ≥ 3, if m0 + ω + 2αγ 2 ≥ 0, where m0 = u0 − α 2 u0xx , then there exists a global solution to Eq. (1.3) in C [0, ∞) ; H s (R) ∩ C 1 [0, ∞) ; H s−1 (R) . To prove Theorem 2.4, it is necessary to have the following lemma. Lemma 2.5. Let u0 ∈ H s (R) , s ≥ 3, such that m0 + ω + 1 − α 2 ∂x2 u0 . Then there is a constant K (> 0) such that
γ 2α 2
≥ 0, where m0 =
|∂x u|∞ ≤ K. Remark. If 0 < γ ≤ M, then the constant K depends only on M. In this case, |∂x u|∞ is uniformly bounded independent of γ . e−| α | , x ∈ R, one has 1 x ∞ −y 1 −x x y e α m (t, y) dy + e α m (t, y) dy. u=p∗m= e α eα 2α 2α −∞ x
Proof. Using p (x) =
1 2α
x
Taking the derivative for u (t, x) with respect to x yields ∞ y x 1 −x x y 1 α α α ∂x u (t, x) = − 2 e e m (t, y) dy + e e− α m (t, y) dy 2 2α 2α −∞ x
x ∞ y y x x 1 −α − e α m (t, y) dy + e α e α m (t, y) dy =− 2 e 2α −∞ x ∞ y 1 x + 2 eα e− α m (t, y) dy α x γ 1 1 x ∞ −y γ 1 α ω+ 2 e α m + ω + 2 dy − = − u (t, x) + 2 e α α 2α α 2α x 1 1 γ ≥ − u (t, x) − ω+ 2 . α α 2α On the other hand, we have x ∞ y y x x 1 1 α ∂x u (x, t) = − 2 e− α e α m (t, y) dy + e e− α m (t, y) dy 2 2α 2α −∞ x 1 γ 1 1 −x x y γ = u (t, x) − 2 e α ω+ 2 e α m + ω + 2 dy + α α 2α α 2α ∞ 1 1 γ ≤ u (t, x) + ω+ 2 . α α 2α As u (t, x) is uniformly bounded independent of γ according to the invariant E(u), there exists a constant K > 0 such that |∂x u|∞ ≤ K, which is the advertised result. Proof of Theorem 2.4. The proof is divided into the following three steps.
Well-Posedness Problem and Scattering Problem for the DGH Equation
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Step 1. Let us claim that if m0 + ω + 2αγ 2 ≥ 0, then m (t) + ω + 2αγ 2 ≥ 0, as long as m (t) exists. In fact, assume u (t, x), t ∈ [0, T ), is a solution of Eq. (1.3). Consider the equation
ξt (t, x) = u (t, ξ (t, x)) − ξ (0, x) = x, x ∈ R.
γ , α2
t ≥0
,
x ∈ R,
(2.12)
In accordance with the Sobolev imbedding theorem and the property of u (t, x), one can see that u (t, ξ ) satisfies Lipschitz condition. And combining with ordinary differential equation theory, we see that there exists the unique solution ξ (t, x) of Eq. (2.12) in C ([0, T )) for any real x. Differentiating Eq. (2.12) with respect to x, we have
ξtx (t, x) = ux (t, ξ (t, x)) ξx (t, x) , x ∈ R. ξx (0, x) = 1,
t ≥0
,
x ∈ R,
(2.13)
From the above Cauchy problem (2.13), we obtain
ξx (t, x) = exp
t
ux (z, ξ (z, x)) dz > 0.
0
In view of the boundedness of ux (t, x) on [0, t0 ] × R for every t0 in [0, T )(according to Lemma 2.5), we see that there exists K (t) such that e−K(t) ≤ ξx (t, x) ≤ eK(t) . So the following two properties of ξ (t, x) are true. (a) The function ξ (t, x) is an increasing diffeomorphism of R of class H 3 (R) with respect to x. (b) lim ξ (t, x) = ±∞, t ∈ [0, t0 ]. x→±∞
Combining with Eq. (1.2), we obtain γ m (t, ξ (t, x)) + ω + 2 ξx2 (t, x) 2α γ = 2 m + ω + 2 ξxt ξx + (mt + mx ξt ) ξx2 2α γ γ = ξx 2 m + ω + 2 ux ξx + ξx mt + mx u − 2 2α α = ξx2 (mt + c0 ux + umx + 2mux + γ uxxx ) = 0. ∂t
As ξ (0, x) = x, x ∈ R, so
m (t, ξ (t, x)) + ω +
γ 2 γ ξx (t, x) = m0 + ω + 2 2 2α 2α
f or all t ≥ 0. Therefore, if the initial potential satisfies m0 + ω + 2αγ 2 ≥ 0, then this inequality will hold under the flow of (1.3), m + ω + 2αγ 2 ≥ 0, for every t ∈ [0, T ).
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Step 2. Now let us obtain the a priori bound for the solution m (t) in L2 . Indeed, by Lemma 2.5 and (2.5), one has d γ |m|22 + 3 ux m2 dx + 4 ω + 2 mux dx = 0. (2.14) dt 2α R R And combining with the following formula
2 2 mux dx ≤ 1 m dx + u dx ≤ K m2 dx, 2 x 2 R R R R we can get that d |m|22 ≤ dt
3K + 4ω +
2γ α2
R
m2 dx ≤ K3 |m|22 ,
where the constant K2 and K3 can be chosen depending only on M when 0 < γ ≤ M. So |m (t)|22 ≤ |m0 |22 etK3 , f or all t ∈ [0, T )
(2.15)
by means of Gronwall’s inequality. Step 3. We are now going to provide an L2 -bounded for mx . Considering the lack of strong smoothing, we shall approximate m0 in H 1 by functions mn0 ∈ H 2 . Moreover, we denote mn = mn •, mn0 for the solution of Eq. (1.2) with initial data mn0 , n ∈ N . In fact, if ρ (x) ∈ C0∞ (R), ρ (x) > 0, R ρ (x) dx = 1, and let ρn (x) = nρ (nx), n m0 = ρn ∗ m0 , n ≥ 1, n ∈ N , then, similar to the a priori estimates (2.15) in Step 2, we can obtain that n m (t, •) 2 ≤ mn (t, •) 2 exp K n t , 0 2 2 where K n := α1 mn0 2 + α12 ω + 2αγ 2 . Differentiating the following equation mnt + 2 ω unx + u mnx + 2 m unx = −γ unxxx with respect to x, multiplying it with mnx and integrating over the whole space. Yield n 2 n 2 n n 2 n d 2 m dx = −5 mx ux dx + 2 m ux dx. dt R x α R R Combining with (2.15), we have n 2 n d mn 2 = −5 mx ux dx 1 dt R
2 n 2 n 2 2 + − 3 u dx ≤ + 8 K n mn 1 . m x 2 2 α α R By using Gronwall’s inequality, we get the result that
n 1 n m ≤ mn exp + 4 K t . 0 1 1 α2
Well-Posedness Problem and Scattering Problem for the DGH Equation
677
Since lim mn0 (x) = m0 (x) in H 1 (R), and the solution m (t, x) depends continun→∞
ously on the initial data m0 , then there exists n(∈ N , enough large), such that n m (t, •) − m (t, •) ≤ 1, t ∈ [0, t0 ]. 1 So
m (t, •)1 ≤ 1 + m0 1 exp
1 n + 4 K t . α2
−1 m (x, t) is a solution of DGH equation, then Considering u (t, x) = 1 − α 2 ∂x2
3 2 u (t, •)3 ≤ 3α + 2 m (t, •)1 α
3 1 2 n 1 + m0 1 exp ≤ 3α + 2 +4 K t , α α2 the proof of Theorem 2.4 is complete. Remark. The proof of Theorem 2.4 shows that, under the above condition of the positivity property, the initial value problem (1.3) has a unique global solution. 3. Weak and Strong Limit as γ → 0 We now turn to the study of the behavior of solutions of the Cauchy problem for (1.3) as the dispersive parameter γ tends to zero. Consider the initial-value problem (1.3) and the analogous problem for the CH equation with the same initial condition, namely t > 0, x ∈ R, ut − α 2 uxxt + 2ω ux + 3uux = α 2 (2ux uxx + uuxxx ) , (3.1) u (0, x) = u0 (x) . For convenience, we always assume α = 1. Theorem 3.1. Under the assumption m0 + ω ≥ 0 and γ > 0 , if u = uγ1 and u = uγ2 are the solutions of the problem (1.3) in C ([0, T ) , H s ) , s ≥ 3, with γ = γ1 and γ = γ2 respectively. Then | u − v |2 converges to zero as γ1 , γ2 → 0. To prove Theorem 3.1, we need the following lemma. s/2 Lemma 3.2. (Kato-Ponce [33]). Let s = 1 − ∂x2 . If s ≥ 0, 1 < p < ∞; f, g ∈ S (R n ), then there exists such a constant c = c (s, n, p) that s , f g ≤ c (|∇f |p s−1 g + s f |g|p ), 1 4 p p3 p2 s s s (f g) ≤ c |f |p g + f |g|p , 1 4 p p p 2
where 1 < p2 , p3 < ∞, and
1 p
=
1 p1
+
1 p2
=
3
1 p3
+
1 p4 .
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Proof of Theorem 3.1. Firstly, we shall prove that the solution u of (1.3) is bounded in H s , s ≥ 3. In fact, we can rewrite Eq. (1.3) as
−1
1 2 2 2 ut + uux + ∂x 1 − ∂x (u + ω) + ux + γ uxx = 0. 2 Since
−1
1 ut + uux + ∂x 1 − ∂x2 (u + ω)2 + u2x + γ uxx 2
−1
1 2 1 γ = ut + u2 + u2x + 2 ω + u u . (3.2) − γ ux − ∂x 1 − ∂x2 x 2 2 2 Equation (3.2) implies that 1 2 + fγ (u) , (3.3) u x 2 −1 2 1 2 where fγ (u) = −∂x 1 − ∂x2 u + 2 ux + 2 ω + γ2 u . Taking the scalar product of Eq. (3.3) with u we get ut − γ ux = −
d u (t)2s = u, ut s = − u, u2 + 2 u, fγ (u) s . x s dt From Lemma 3.2, one can see that u, u2 = 2 u, uux s = 2 s (uux ) , s u x s = 2 us ux , s u + s , u ux , s u ≤ c |∂x u|∞ u2s + c s , u ux 2 s u 2 2 ≤ c |∂x u|∞ u2s + c |∂x u|∞ s u 2 2 +c |∂x u|∞ s u 2 ≤ cs |ux |∞ u2s ,
(3.4)
(3.5)
where in the last estimate above we used the result of the uniform boundedness of ∂x u in Lemma 2.5 and the constant c0 depends only on the bound M of γ , but not γ . On the other hand, from the Cauchy-Schwarz inequality, we have u, fγ (u) ≤ us fγ (u) . s s According to Lemma 2.5, we estimate fγ (u)s as follows: fγ (u) ≤ u2 + 2 ω + γ u + 1 u2 x s 2 2 s−1
≤ c1 u2 + us−1 + u2x s−1 s−1 ≤ c1 |u|∞ us−1 + us + |∂x u|∞ ∂x us−1 ≤ c1 (u1 + 1 + |∂x u|∞ ) us ≤ c0 us ,
(3.6)
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where c0 , c1 are constants which depend only on M with |γ | ≤ M. So combining (3.5) with (3.6), one can deduce from (3.4) that d u (t)2s ≤ c0 u2s . dt And by using Gronwall’s inequality, we get that the solution u of (1.3) is bounded in H s , s ≥ 3, for any t ∈ [0, T ], where c0 is independent of γ , but M with all |γ | ≤ M. Next, we show that the sequence of solutions of (1.3) is a Cauchy sequence in L2 (R). Let u = uγ1 and v = uγ2 be the solutions of problem (1.3) with γ = γ1 and γ = γ2 respectively. Then the function w = u − v satisfies wt + wux + wx v = γ1 wx + (γ1 − γ2 ) vx + fγ1 (u) − fγ2 (v) .
(3.7)
Considering fγ1 (u) − fγ2 (v) −1 1 2 w (u + v) + wx (ux + vx ) + 2 ω w + (γ1 − γ2 ) u + γ2 w , = −∂x 1 − ∂x 2 one can see that fγ (u) − fγ (u) ≤ w (u + v) + 1 wx (ux + vx ) +2 ω w + (γ1 − γ2 ) u + γ2 w 1 2 s 2 s−1 1 ≤ w (u + v)s−1 + wx (ux + vx )s−1 2 +2 ω ws−1 + |γ1 − γ2 | us−1 + |γ2 | ws−1 1 ≤ ws u + vs + ws u + vs + 2 ω ws + |γ1 − γ2 | us + |γ2 | ws 2
3 ≤ ws (3.8) (us + vs ) + 2 ω + |γ2 | + |γ1 − γ2 | us . 2 Multiplying (3.7) with w and integrating on R with respect to x, and integration by parts, we obtain d |w|2L2 = − w 2 (2ux + vx ) dx + (γ1 − γ2 ) wvx dx dt R R + w fγ1 (u) − fγ2 (v) dx R ≤ (2 |ux |∞ + |vx |∞ ) |w|2L2 + |γ1 − γ2 | |u|L2 + |v|L2 |vx |L2 + fγ1 (u) − fγ2 (v) L2 |w|L2 ≤ (2 us + vs ) |w|2L2 + |γ1 − γ2 | (us + vs ) vx s + us
3 + (us + vs ) + 2 ω + |γ2 | |w|2L2 2 ≤ 2 (2 us + vs + 2 ω + |γ2 |) |w|2L2 + |γ1 − γ2 | (us + vs ) vx s + us by using (3.8). Integrating this inequality over [0, T ] for any T > 0 and applying Gronwall’s inequality yield
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|w|2L2 ≤ cT |γ1 − γ2 | K u (t)s , v (t)s , ∀ t ∈ [0, T ]. Since K u (t)s , v (t)s is uniformly bounded and independent of γ , |w|L2 converges to zero as γ1 , γ2 → 0 and uγ is a Cauchy sequence in L2 as γ → 0, uniformly with respect to t ∈ [0, T ). This completes the proof of Theorem 3.1. By Theorem 3.1 and the a priori estimates established for solutions of the DGH equation independent of γ , we can obtain that the Cauchy sequence of solutions of the DGH equation locally strongly converges to the solution of the CH equation as γ tends to zero. Corollary 3.3. Under the assumption m0 + ω ≥ 0,and γ > 0 let uγ be the solution of (1.3) in H s , s ≥ 3. Then uγ converges to the solution of (3.1) in H s , s ≥ 3 as γ → 0. Proof. First of all, according to Theorem 3.1, and letting u = l im uγ in L2 , where uγ γ →0
is a Cauchy sequence in L2 as γ → 0 uniformly with respect to t ∈ [0, T ], we will show that u is the solution of (3.1). Indeed, since uγ is the solution of (1.3) in H s , s ≥ 3, for t ∈ [0, T ), we have
t 1 2 dτ, (3.9) Sγ (t − τ ) fγ (u) − u uγ (t) = Sγ (t) u0 + x 2 0 1 where Sγ (t) v = 2π exp i (ξ x − γ ξ t) vˆ (ξ ) dξ and Sγ (t) satisfies the relation R Sγ (t) v = vs with s ≥ 0 and v ∈ H s . Since Sγ (t) u0 ≤ 1 ˆ 0 (ξ ) dξ ≤ 2π R u s 1 3 2π u0 s , for s ≥ 2 , and from Lebesgue’s dominated convergence theorem we can get that Sγ (t) u0 → S (t) u0 , 0 ≤ t < T , as γ → 0. 1 where S (t) u0 = 2π ˆ 0 (ξ ) dξ . R exp (i ξ x) u On the other hand, for s ≥ 3, t ∈ [0, T ), and τ ∈ [0, t], it follows from (3.6) that
S (t − τ ) − 1 u2 ≤ S (t − τ ) − 1 u2 2 γ x 2 γ x s−1 2 1 2 ≤ ≤ u2γ ≤ uγ s , uγ x s−1 s 4π S (t − τ ) fγ (u) ≤ S (t − τ ) fγ (u) ≤ 1 fγ (u) ≤ c0 us . s s 2π So the right-hand side of (3.9) is bounded uniformly independent of γ . Therefore, by using Lemma 2.5, and in view of the Lebesgue dominated convergence theorem, passing to the limit as γ → 0 in (3.9), we get that
t 1 2 u dτ, S (t − τ ) f (u) − u (t) = S (t) u0 + x 2 0 where f (u) := f0 (u). Hence u ∈ L∞ 0, T ; L2 satisfies (3.1). The local existence for the CH equation implies that equation (3.1) has a unique solution in C ([0, T ); H s ), s ≥ 3. This proves that u is the strong solution of (3.1). This completes the proof of Corollary 3.3.
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4. The Behavior of Solutions as α → 0 In this section, we shall demonstrate the behavior of solutions of the Cauchy problem (1.3) as the parameter α 2 tends to zero. For convenience, we shall always replace α 2 with ε, 2ω with k. So Eq. (1.3) is rewritten as follows: ut − ε uxxt + k ux + 3uux + γ uxxx = ε (2ux uxx + uuxxx ) x ∈ R, t ∈ R. u (x, 0) = u0 (x) , (4.1) 4.1. A priori estimates. For later estimation of Sobolev norms of solutions, we will require the following basic inequalities, due to Y.A. Li and P.J. Olver [36]. Lemma 4.1.1. Given q ≥ 0, let u = u (x) ∈ H q be any function such thatux L∞ < ∞. Then there is a constant cq depending only on q such that the following inequalities hold: ∧q u ∧q (uux ) dx ≤ cq ux L∞ u2 q , (4.2) H R ∧q u ∧q u2 dx ≤ cq uL∞ u2 q . (4.3) H R
Moreover, if u and f are functions in H q+1 ∩ {ux L∞ < ∞}, then q ∈ 21 , 1 , cq f H q+1 u2H q , q u q (uf )x dx ≤ cq f H q+1 uL∞ uH q + f H q ux L∞ uH q R q ∈ (1, ∞) . + fx ∞ u2 q , L
H
(4.4) Theorem 4.1.2. Suppose that for some s ≥ 3, the function u (x, t) is a solution of Eq. (4.1) corresponding to the initial data u0 ∈ H s .Then for any real number q ∈ (0, s], there exists a constant c depending only on q, such that u2H q + ε ux 2H q ≤ u0 2H q + ε u0x 2H q
t q 2 2 ux L∞ u + ε q ux dx dt. +c 0
(4.5)
R
For any q ∈ (1/2, s] and any r ∈ (1/2, q], there exists a constant c depending only on r and q, such that u2H q + ε ux 2H q ≤ u0 2H q + ε u0x 2H q
t 2 2 1/2 2 2 r+1 u + ε r+1 ux dx q u + ε q ux dx dt. +c 0
R
R
(4.6)
Moreover, for any q ∈ (0, s], there exists a constant c depending only on q, such that ut H q + ε ut H q+1 ≤ c uH q+3 c (k, γ ) + uH 1+β , (4.7) for all β > 0. And under the assumption γ = O (ε), one can see that ut H q + ε ut H q+1 ≤ c uH q+2 c (k) + uH 1+β .
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Proof. For any q ∈ (0, s], applying (q u) q to both sides of Eq. (4.1) and integrating with respect to x again, one obtains the equation
2 2 q u + ε q ux dx R ε q q = −3 u (uux ) dx + q u q ∂x3 u2 + q ux q u2x dx 2 R R = (−3 + ε) q u q (uux ) dx R ε q+1 q+1 −ε u q ux q u2x dx (uux ) dx + 2 R R
1 d 2 dt
using integration by parts. It follows from the inequalities (4.2) and (4.3) that there is a constant cq such that 1 d 3 u2H q + ε ux 2H q ≤ 4cq ux L∞ u2H q + εcq ux L∞ ux 2H q 2 dt 2 ≤ 4cq ux L∞ u2H q + ε ux 2H q .
Integrating with respect to t on both sides of the above inequality leads to inequality (4.5). Applying the inequality
ux L∞ ≤ cr+1 uH r+1 ≤ cr+1
r+1
2 u
2 1/2 r+1 +ε ux dx
R
for r > 21 to the right-hand side of (4.5) yields the estimate (4.6). For any q ∈ (0, s], applying (q ut ) q to both sides of Eq. (4.1) and integrating with respect to x again, one obtains the equation
2 2 q ut + ε q uxt dx R q q q q = −3 ut (uux ) dx−k ut (ux ) dx−γ q ut q (uxxx ) dx R R R ε + q ut q ∂x3 u2 − q ut q u2x dx x 2 R = (−3 + ε) q ut q (uux ) dx R ε q+1 q+1 −ε ut q uxt q u2x dx (uux ) dx + 2 R R −k (4.8) q ut q (ux ) dx − γ q ut q (uxxx ) dx.
1 d 2 dt
R
R
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By using the inequalities (4.2) (4.3), one may obtain the following inequalities: q ut q (uux ) dx ≤ cq ut H q uˆ 1 uH q+1 ≤ cq ut H q uH 1 uH q+1 , L R ε q+1 ut q+1 (uux ) dx ≤ cq ε ut H q+1 uˆ L1 uH q+2 R
≤ cq ε ut H q+1 uH 1 uH q+2 , q q 2 ε ut ux dx ≤ cq ε uxt H q uˆ x L1 ux H q x R
≤ cq ε uxt H q uH 1+β ux H q ≤ cq ε ut H q+1 uH 1+β uH q+1 (∀β > 0), ε q q uxt u2x dx ≤ cq ε uxt H q uH 1+β ux H q (∀β > 0), 2 R q ut q (ux ) dx ≤ cq ut H q ux H q , R q q γ ut (uxxx ) dx ≤ cq γ ut H q uxxx H q . R
Applying the above six inequalities to (4.8) yields the inequality ut 2H q + ε uxt 2H q ≤ c (ut H q + ε uxt H q ) uH q+3 k + γ + uH 1+β , where c = c (q) is constant independent of ε, i.e.,
ut H q + ε ut H q+1 ≤ c uH q+3 k + γ + uH 1+β
or
ut H q + ε ut H q+1 ≤ c uH q+2 k + uH 1+β
with γ = O (ε). So we have
ut H q + ε ut H q+1 ≤ c uH q+3 k + γ + uH 1+β , f or all β > 0.
And under the assumption γ = O (ε), one can see that ut H q + ε ut H q+1 ≤ c uH q+2 k + uH 1+β for some constant c independent of ε. Lemma 4.1.3. Under the above assumptions, the following estimates hold for any ε with 0 < ε < 1/8 uε0 H q ≤ c, (q ≤ s), uε0 H q ≤ cε
s−q 8
, (q > s), s−q 8
uε0 − u0 H q ≤ cε , (q ≤ s), uε0 − u0 H s = o (1) .
(4.9) (4.10)
(4.11) (4.12) Here uε0 = φε ∗u0 is the convolution uε0 of the functions φε (x) = ε−1/8 φ ε−1/8 x and u0 such that the Fourier transform φˆ of φ satisfies φˆ ∈ Cc∞ ([0, ∞) ; H ∞ ) , φˆ (ξ ) ≥ 0, and φˆ (ξ ) = 1 for any ξ ∈ (−1, 1) .
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Lemma 4.1.4. There exist constants c1 and c2 independent of ε such that the following inequalities hold 4Mr
u2H r + ε ux 2H r ≤
M
≤
1/2 2 2 − cMr t
3/2 < r ≤ s,
,
(2 − Mt)2
(4.13)
1/2 where Mr = u0 2H r + ε u0x 2H r , M = max 4Mr , cMr .,
u2H s + ε ux 2H s
1/2
u2H s+p + ε ux 2H s+p
c1 , (2 − Mt)c2
≤
1/2
ut 2H s+p + ε uxt 2H s+p
(4.14)
p
≤
1/2
c1 ε − 8 , p > 0, (2 − Mt)c2
(4.15)
p+1
≤
c1 ε − 8 , p > −1 (2 − Mt)c2
(4.16)
for any ε sufficiently small. Proof. Choose a fixed number r with 3/2 < r ≤ s. It follows from (4.6) that r 2 2 r 2 2 u + ε r ux dx ≤ u0 + ε r u0x dx R
R
+c
t 0
Multiplying (4.17) with
2
r
u
+ ε ux r
3/2
·
3/2 2 + ε r ux dx dt. (4.17)
R
≤
dx
2
r
1/2
, we obtain the inequality
+ ε ux
u
R
2
r u
(r u)2 + ε (r ux )2 dx
R
2
r
2
1/2 dx
R
r
u0
2
2
+ ε u0x r
dx + c
t
r
2
u
0
R
+ ε ux r
2
3/2 dx
dt .
R
(4.18) Denote that
3/2 t r 2 2 r 2 r r 2 V (t) := u0 + ε u0x dx + c u + ε ux dx dt, 0
R
R
and we have d V (t) ≤ dt
2
r
u
+ ε ux r
2
1/2 dx
V (t)
R
from the inequality (4.17). It follows from Gronwall’s inequality and (4.17) that 4Mr
u2H r + ε ux 2H r ≤
1/2
2 − cMr t
2 ≤
M (2 − Mt)2
,
3/2 < r ≤ s
(4.19)
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685
holds for any t ∈ [0, 2/M). Substituting the inequality (4.13) into (4.6) with q = s and u = uε , one obtains the estimate u2H s + ε ux 2H s ≤ u0 2H s + ε u0x 2H s
t √ s 2 s 2 M u + ε ux dx dt +c R 0 (2 − Mt) for any t ∈ [0, 2/M). It follows from Gronwall’s inequality and (4.9), (4.10) that there are constants c1 , c2 depending on M such that 1/2 c1 u2H s + ε ux 2H s ≤ . (2 − Mt)c2 According to the inequalities (4.9), (4.10), one can see that Ms = u0 2H s + ε u0x 2H s ≤ c0 1 + ε 3/4 ≤ c, where c0 , c is independent of ε. So M, c1 , one may obtain
u2H s+p + ε ux 2H s+p
1/2
c2 are independent of ε. In a similar way, p
≤
c1 ε − 8 , p > 0. (2 − Mt)c2
And by using the inequalities (4.7), (4.15), we obtain the inequality
ut 2H s+p
+ ε uxt 2H s+p
1/2
p+1
c1 ε − 8 ≤ , p > −1. (2 − Mt)c2
Remark. According to the inequality (4.14) in Lemma 4.1.4, one can see that uH s is uniformly bounded and the boundedness is independent of ε. 4.2. The convergence of solutions as ε → 0. We shall demonstrate that {uε } is a Cauchy sequence. We rewrite Eq. (4.1) as follows in this section: ut − ε uxxt + k ux + (3 − ε) uux + γ uxxx
1 = −ε 1 − ∂x2 (uux ) − ε ∂x u2 + u2x . 2 Let uε and uδ be solutions of (4.1), corresponding to the parameters ε and δ, respectively, and let w = uε − v δ and f = uε + v δ . Then w satisfies the problem 1 (3 − ε) (wf )x 2 + (δ − ε) vvx + γε wxxx + (γε − γδ ) vxxx ε = − 1 − ∂x2 (wf )x + (δ − ε) 1 − ∂x2 (vvx ) 2
1 1 2 2 −ε ∂x wf + wx fx + (δ − ε) ∂x v + vx . 2 2
wt − ε wxxt + (δ − ε) vxxt + k wx +
Without loss of generality, we always assume that 0 < ε ≤ δ ≤ 2ε ≤ 1/8 .
(4.20)
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Theorem 4.2.1. Under the assumption γ = O (ε), there exists T > 0, independent of ε, such that {uε } and {uεt }are Cauchy sequence in the space C ([0, T ) ; H s (R)) and C [0, T ) ; H s−1 (R) , respectively. Proof. First, for a constant q with 1/2 < q ≤ min {1, s − 1}, multiplying by 2q w on both sides of Eq. (4.1) and integrating with respect to x again, and we may obtain the equation q 2 2 1 d w + ε q wx dx 2 dt R q q = − (δ − ε) w vxxt dx R q q q q 1 −k w wx dx− (3 − ε) w (wf )x dx 2 R R q q q q − (δ − ε) w (vvx ) dx − γ w (wxxx ) dx R R q q − (γε − γδ ) w (vxxx ) dx R q q+2 q q+2 ε w w − (wf )x dx + (δ − ε) (vvx ) dx 2 R R q q −ε w (wf )x dx R q q q q 2 1 − ε dx w (wx fx )x dx + (δ − ε) w v x 2 R R
q 1 2 + (δ − ε) dx w q vx 2 R x q q q q 1 = − (δ − ε) w vxxt dx − (3 − ε) w (wf )x dx 2 R R q q q q w (vvx ) dx − (γε − γδ ) w (vxxx ) dx − (δ − ε) R R q q+2 q q+2 ε − w w (wf )x dx + (δ − ε) (vvx ) dx 2 R R q q q q 1 −ε w (wf )x dx − ε w (wx fx )x dx 2 R R
q q 2 q 1 2 q + (δ − ε) dx. dx + (δ − ε) w v w v x 2 x x R R Consider the following inequalities: q q (δ − ε) w vxxt dx ≤ δ vt H q+2 wH q , R q q w (wf )x dx ≤ c3 (wx L∞ + vH q ) w2H q R +c3 vH q+1 + wH q wH q ,
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q q (δ − ε) ≤ δ vH q vH q+1 wH q , w dx (vv ) x R q q (γε − γδ ) ≤ cδ vH q+3 wH q , w dx (v ) xxx R ε q w q+2 (wf )x dx 2 R ≤ c3 wH q ε f H q+3 f H q+3 + vH q+3 ≤ c3 ε 1/4 wH q , q q+2 (δ − ε) w dx ≤ δ vH q+2 vH q+3 wH q , (vv ) x R q q ε ≤ ε 2 vH q+1 f H q+1 wH q , dx w (wf ) x R ε q q ≤ c4 ε f H q+1 w2 q+1 , w f dx (w ) x x x 2 H R q q 2 (δ − ε) dx ≤ δ v2H q+1 wH q , w v x R
q 1 2 q (δ − ε) dx ≤ δ v2H q+2 wH q , w v x 2 R x and by using the inequalities (4.14), (4.15), one can see that d dt
q w
2
R
2 + ε q wx dx ≤ c δ ρ wH q + δ 1/4 wH q + w2H q ,
1, s ≥ 3 + q . Combining with inequality (4.10), we obtain the 1+s−q 4 , s <3+q following inequality: where ρ =
wH q ≤ c1 δ
s−q 4
ect + δ ρ ect − 1 + δ 1/4 ect − 1
.
In a similar way, combining with the inequalities (4.9) and (4.10), one can obtain 3 wH s ≤ c1 w0 H s + δ 4 ect + δ m ect − 1 where m = min
1 s−q−1 4, 4
,
and
wH s+p ≤ c1 δ for all p > 0. So wH s → 0, as ε,
−p 4
ect + δ
δ → 0.
1−p 4
ect − 1 ,
(4.21)
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Next, we consider convergence of the sequence {uεt }. Multiplying by 2s−2 wt on both sides of Eq. (4.20) and integrating with respect to x again, then we may obtain the equation 2 2 1 d s−1 wt + ε s−1 wxt dx 2 dt R s−1 s−1 s−1 wt s−1 wx dx = − (δ − ε) wt vxxt dx − k R R 1 s−1 s−1 − (3 − ε) wt (wf )x dx 2 R − (δ − ε) s−1 wt s−1 (vvx ) dx R s−1 − s−1 wt s−1 (vxxx ) dx wt s−1 (wxxx ) dx− (γε −γδ ) R R ε s−1 s+1 − s−1 wt s+1 (vvx ) dx wt (wf )x dx+ (δ − ε) 2 R R 1 −ε s−1 wt s−1 (wf )x dx − ε s−1 wt s−1 (wx fx )x dx 2 R R + (δ − ε) s−1 wt s−1 v 2 dx x R
1 2 + (δ − ε) s−1 wt s−1 dx. vx 2 R x By using the following inequalities: s−1 s−1 (δ − ε) wt vxxt dx ≤ δ vxxt H s−1 wt H s−1 , R s−1 s−1 k wt wx dx ≤ k wx H s−1 wt H s−1 , R 1 s−1 s−1 2 (3−ε) w wt H s−1 , v w s s dx ≤c w + (wf ) s t H H x H 2 R s−1 s−1 (δ − ε) ≤ δv2 s wt H s−1 , w dx (vv ) t x H R γε s−1 wt s−1 (wxxx ) dx ≤ γε wH s+2 wt H s−1 , R s−1 s−1 (γε − γδ ) wt (vxxx ) dx ≤ γδ vH s+2 wt H s−1 , R ε s−1 s+1 ≤ c ε1/2 wH s+2 ε 1/2 f H s+2 wt H s−1 , dx w (wf ) t x 2 R and combining with (4.9), (4.10), (4.14), (4.15), (4.21), we have wt H s−1 ≤ c δ 1/2 wH s+2 + δ 1/2 → 0, as ε,
δ → 0.
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689
From Theorem 4.2.1, one can see that Corollary 4.2.2. Under the assumption ε > 0 and γ = O(ε), let uε be the solution of (4.1) with the initial data uε0 = φε ∗ u0 in H s , s ≥ 3. Then uε converges to the solution of the following Eq. (4.22) with the initial data u0 in H s , s ≥ 3 as ε → 0, ut + kux + 3uux = 0, x ∈ R, t ∈ R. (4.22) u (x, 0) = u0 (x) , Moreover, from the proof of Theorem 4.2.1, one can see that without the assumptionγ = O (ε), we have Theorem 4.2.3. If u = uε and v = uδ are the solutions of the problem (4.1) in C ([0, T ), H s ) , s ≥ 3, with ε and ε = δ respectively. Then u − vH s−3 converges to zero as ε, δ → 0. 5. Stability of Solitary Waves Let us now discuss the appropriate notion of stability for the solitary waves of (1.3). By a solitary wave solution of Eq. (1.3), we mean a traveling wave solution of the form u (x, t) = ϕ (x − ct), where c > 0 is the wave speed and ϕ is a solution of the following stationary problem: −cϕ + cα 2 ϕxx + 2ω ϕ + 23 ϕ 2 + γ ϕxx = α 2 ϕϕxx + 21 ϕx2 , x ∈ R, (5.1) ϕ ∈ H 1 , ϕ = 0, where the constant γ > 0. Multiplying both sides of (5.1) by ϕx and integrating again, we get γ α 2 ϕx2 c + 2 − ϕ = ϕ 2 (c − 2ω − ϕ) . (5.2) α From (5.2) we infer that solitary waves exist only for c > 2ω and each such speed c determines ϕ uniquely up to translations (see [4, 5, 17]). Moreover the solitary wave ϕ is smooth and positive with the peak of height c − 2ω. Furthermore, 2+γ 1 2ωα |x| , |x| → ∞, ϕ (x) = O exp − 1− α cα 2 + γ as ϕx2 ≈ 1 −
2ωα 2 +γ cα 2 +γ
, for |x| → ∞. By (5.2), we get an implicit expression of the
solitary wave and its smooth dependence on the constant c, α, γ , ω. As 2ωα 2 + γ → 0, the solitary waves of (1.3) with maximal elevation at x = 0 converge uniformly on 1 every compact subset of R to the peakon Ce− α |x| (see [16]). In section, we will give the peakon solution in condition 2ωα 2 + γ = 0. Definition 5.1. The solitary wave ϕ of (1.3) every ε > 0 there is δ > 0 is stable if for such that if 0 < T ≤ ∞ and u ∈ C [0, T ), H 1 (R) is a solution to (1.3) with u (0) − ϕH 1 (R) ≤ δ, then for every t ∈ [0, T ), inf u (t, •) − ϕ (• − ξ )H 1 ≤ ε.
ξ ∈R
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Now let us prove the following Theorem 5.2 about stability of solitary waves. Theorem 5.2. All solitary waves of (1.3) are stable. Proof. Following [27], a spectral analysis of the linearized Hamiltonian operator reduces the question of the orbital stability to the question of whether or not a certain modified energy is a convex function of the wave speed. In terms of the functions E and F , E (ϕ) = ϕ − α 2 ϕxx ,
F (ϕ) =
3 2 1 ϕ + 2ωϕ + γ ϕxx − α 2 ϕϕxx − α 2 ϕx2 , 2 2
and according to Eq. (5.1), we can easily obtain that cE (ϕ) − F (ϕ) = 0,
(5.3)
F respectively in H 1 (R). where E and F are the Frechet derivatives of E and The linearized Harmiltonian operator Hc of cE − F around ϕ is given by Hc = cE (ϕ) − F (ϕ) γ = α 2 2ϕ − 2 c + 2 ∂x2 + 2α 2 ϕ ∂x + 2α 2 ϕ − 6ϕ + 2 (c − 2ω) α γ 2 = −α ∂x 2 c + 2 − 2ϕ ∂x + 2α 2 ϕ − 6ϕ + 2 (c − 2ω) . α Since ϕ, ϕx , ϕxx → 0 exponentially fast as |x| → ∞, γ γ 2 c + 2 − 2ϕ (x) ≥ 2 ω + 2 > 0 α α on R. It follows that the spectral equation Hc v = ηv can be transformed by the Liouville substitution x 1 1 γ 4 v (x) − 2ϕ z= dx, ψ = 2 c + " (x) (z) 2 α 0 γ 2 c + α 2 − 2ϕ (x) into Lc ψ (z) = −∂z2 + qc (z) + qc (z) =
2c−4ω α2
ψ (z) =
η ψ α2
(z), where
3 ϕ 2 (x) −6 . x ϕ (x) + 2 ϕxx − 2 α 2α 8α 2 c + αγ2 − ϕ
From (5.2), we obtain that qc (z) =
−6 ϕ (x) + 2α3 2 ϕxx α2
−
ϕ(c−2ω−ϕ) , then qc → 0 8α 4 c+ γ2 −ϕ α : H 1 (R) → H −1 (R) is
exponentially as |z| → ∞, which gives the result that Lc a self-adjoint operator with essential spectrum 2c−4ω , ∞ . So we may have finitely α2
many eigenvalues of Lc located to the left of 2c − 4ω > 0. The n th eigenvalues (in increasing order) have up to a constant multiple, a unique eigenfunction with precisely (n−1) zeros. For these matters we refer to [32]; the Liouville transformation ensures that the same spectral information holds for the differentials operator Hc mapping H 1 (R) into H −1 (R). Note that (5.3) means Hc (ϕx ) = 0. The behavior of the function ϕ tells
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us that ϕx has exactly one zero. So the zero eigenvalue of the operator Hc is simple and there is exactly one negative eigenvalue while the rest of the spectrum is positive and bounded away from zero. As for the above results, it is known [27] that stability would be ensured by the convexity of the scalar function dc = cE (ϕ) − F (ϕ) , c > 2ω. Differentiating with respect to c and taking that into account, we find # $ ∂ϕc d (c) = cE (ϕ) − F (ϕ) , + E (ϕ) = E (ϕ) , c > 2ω. ∂c From (5.2), we have (c − ϕ − 2ω) α 2 c − α 2 ϕ + γ > 0. Equation (5.3) then follows (5.3) and the fact that ϕ is even:
∞ ∞
1 c − 2ω − ϕ 2 2 2 2 2 2 2 E (ϕ) = ϕ +α ϕx dx= ϕ +α ϕx dx = ϕ + 1 dx 2 R α2 c + γ − α2 ϕ 0 0 ∞ 1 + α 2 (c − ϕ) + γ − 2ω =− ϕϕx % dx. 0 (c − ϕ − 2ω) α 2 c − α 2 ϕ + γ The change of variables y = c + αγ2 − ϕ (x) yields γ 2 c+ γ c + γ − y 1 + α + γ − 2ω y − 2 2 2 α α α E (ϕ) = dy, " γ 2ω+ 2 γ α α y y − 2 ω + 2α 2 and the right-hand side is an increasing function of c, as one can check by differentiation. From these we infer that d (c) > 0, then the proof of Theorem 5.2 is complete. 6. The Spectral Problems and the Scattering Theory for the DGH Equation 6.1. Lax pair of the DGH equation. To study the spectral theory of the DGH equation (1.3), we begin with its Hamiltonian structure. Considering that Eq. (1.3) is biHamiltonian, according to (2.1), (2.2), and by the standard Gelfand-Dorfman theory (see [28]), its bi-Hamiltonian property implies that the nonlinear Eq. (1.3) arises as a compatibility condition for two linear equations, namely, the isospectral eigenvalue problem 1 α 2 − 2γ λ ψxx = ψ + λ (m + ω) ψ, (6.1) 4 and the evolution equation for the eigenfunction ψ,
1 1 ψt = − u ψx + ux ψ. (6.2) 2λ 2 Compatibility of these linear equation (ψxxt = ψtxx ) and isospectrality (λ t = 0) λ imply Eq. (1.3). And the spectral parameter transformation η = α 2 −2γ converts (6.1), λ (6.2) into 1 γ ψxx = ψ, (6.3) ψ + η m + ω + 4α 2
2α 2
1 1 1 ψt = (6.4) + 2γ − u ψx + ux ψ. 2 2α η 2
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Now let us point out how one comes up with the isospectral problem associated to (1.4). We consider two operators L and M. L is the following Schrodinger operator of the spectral problem L = −∂x2 + V (x, t, λ) , where λ is a parameter, and M is the operator governing the associated time evolution of the eigenfunction. We look for a solution M of the Lax pair system of Eq. (1.4) to satisfy the folLψ =0 lowing: and we wish to express Eq. (1.4) in the Lax form (Lt + [L , M]) ψ = 0, Lt + [L , M] = 0, where L is a self-adjoint operator, and M is to be an anti-symmetric operator. Now suppose M = a (x, t) ∂x3 + b (x, t) ∂x + c (x, t), so we can find 1 V (x, t, λ) = 4α1 2 + αλ2 (m + ω), and a (x, t) = γ , b (x, t) = 2λ − u, c (x, t) = 1 2 ux + β, where β is an arbitrary constant. Thus Eq. (1.4) can be written like the compatibility condition ψxxt = ψtxx between a second order non-homogeneous spectral problem of Sturm-Liouville type ψxx = 4α1 2 ψ + αλ2 (m + ω) ψ, and a linear evolu1 − u ψx + 21 ux + β ψ. Its isospectral condition is tion equation ψt = γ ψxxx + 2λ λt = 0. 6.2. Wronskian antisymmetric bilinear form. Let us review the wronskian antisymmetric bilinear form about Eq. (1.3). Considering (6.3), and noting k 2 = − 4α1 2 − η ω + 2αγ 2 , under the assumption that m ∈ H 1 (R) satisfies R (1 + |x|) |m (x)| dx < ∞, we get that ψ (x, k) = −k 2 ψ (x, k) + ηmψ (x, k) ,
−ψ ≈ k 2 ψ (|x| → ∞) .
(6.5)
These suggest the introduction of the complex-valued solutions ϕ (x, k), ϕ¯ (x, k), ψ (x, k), ψ¯ (x, k) with the asymptotic behavior ϕ (x, k) ≈ e−ikx , ϕ¯ (x, k) ≈ eikx (x → −∞) ; ψ (x, k) ≈ eikx , ψ¯ (x, k) ≈ e−ikx (x → ∞) . By this boundary condition, we have ϕ (x, k) = ϕ¯ (x, −k) ,
ψ (x, k) = ψ¯ (x, −k) .
(6.6)
As m (x) is a real function, so when k ∈ R, the above relations can be regarded as conjugate relations. Introducing the Wronskian antisymmetric bilinear form, W (ϕ (x, k) , ψ (x, k)) := ϕ (x, k) ψx (x, k) − ϕx (x, k) ψ (x, k) . Taking into account the boundary condition, it can be seen that W (ϕ (x, k) , ϕ¯ (x, k)) = 2ik = −W ψ (x, k) , ψ¯ (x, k) .
(6.7)
So when k = 0, solutions are linearly independent respectively. Since any three solutions of Eq. (6.3) are linearly dependent, we can see that, for every k ∈ R\ {0}, relations of the form ϕ (x, k) = a (k) ψ¯ (x, k) + b (k) ψ (x, k) , ϕ¯ (x, k) = −a¯ (k) ψ (x, k) + b¯ (k) ψ¯ (x, k) .
(6.8)
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Using (6.6) we can easily obtain symmetry conditions as follows: a¯ (k) = −a (−k) = −a ∗ k ∗ , b¯ (k) = b (−k) = b∗ k ∗ ,
(6.9)
where a ∗ (k ∗ ), b∗ (k ∗ ) are the & complex conjugates of'a (k), b (k) respectively. Noting that W (ϕ (x, k) , ϕ¯ (x, k)) = a (k) a¯ (k) + b (k) b¯ (k) W ψ (x, k) , ψ¯ (x, k) , according to (6.8). By (6.7) we obtain that a (k) a¯ (k) + b (k) b¯ (k) = −1, and from (6.9) we have 1 W (ϕ (x, k) , ψ (x, k)) , 2ik 1 b (k) = − W ϕ (x, k) , ψ¯ (x, k) . 2ik
|a (k)|2 − |b (k)|2 = 1, a (k) =
(6.10)
¯ It is more convenient to work with the modified eigenfunctions M (x, k), M(x, k), ¯ N (x, k) and N(x, k), defined by M (x, k) = ϕ (x, k) eikx , M¯ (x, k) = ϕ¯ (x, k) eikx , N (x, k) = ψ (x, k) eikx , N¯ (x, k) = ψ¯ (x, k) eikx , then ¯ M (x, k) ≈1, M(x, k)≈e2ikx , x → −∞, N (x, k) ≈ e2ikx , N¯ (x, k) ≈ 1,
x → +∞,
so we have M (x, k) M¯ (x, k) = N¯ (x, k) + ρ (k) N (x, k) , = −N (x, k) + ρ¯ (k) N¯ (x, k) , a (k) a¯ (k) (6.11) ¯
b(k) b(k) 1 where ρ (x) = a(k) , ρ¯ (k) = a(k) . τ (k) = a(k) and ρ (k) are called transmission and ¯ reflection coefficients respectively. From (6.10) we see that τ (k) and ρ (k) satisfy |ρ (k)|2 + |τ (k)|2 = 1, which means capacity conservation on the elasticity scattering. Noting that N (x, k) = N¯ (x, −k) eikx , therefore from (6.11) we obtain
M (x, k) = N¯ (x, k) + ρ (k) eikx N¯ (x, −k) . a (k) With respect to M (x, k) , a (x, k) , N¯ (x, k), we have the following results: Lemma 6.2.1. (i)M (x, k) and a (k) can be analytically extended to the upper half k-plane; N¯ (x, k) can be analytically extended to the lower half k-plane; they are written respectively as the following : x 2ik(x−y) e −1 M (x, k) = 1 + η m (y) M (y, k) dy , 2ik −∞ ∞ 2ik(x−y) e −1 m (y) N¯ (y, k) dy, N¯ (x, k) = 1 − η 2ik x ∞ 1 m (y) M (y, k) dy , a (k) = 1 + η −∞ 2ik ∞ η b (k) = m (y) M (y, k)e−2ky dy. 2ik −∞ (ii) The zeros of the function a (k) in the upper half-plane lie on the imaginary axis, and all these in the upper half-plane are simple zeros.
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Proof. (i). Arguments similar to Lemma 2.2.1 in [34, 35]. (ii). From |a (k)|2 − |b (k)|2 = 1 (where k ∈ R\ {0}) one can see that there don’t exist zeros of the function a (k) on R\ {0}. Assume that k lies on the upper half-plane, which satisfies a (k) = 0, from (i) and W (ϕ, ψ) = 2ika (k), I m (k) > 0, we can see W (ϕ (x, k) , ψ (x, k)) = 0, so ϕ (x, k) , ψ (x, k) are linearly dependent: ϕ (x, k) = cψ (x, k). As ϕ (x, k) ≈ e−ikx (x → −∞), ψ (x, k) ≈ eikx (x → ∞), then ψ (x, k) ∈ L2 (R);and in view of ψ (x, k) = −k 2 ψ (x, k) + ηmψ (x, k), m (x) ∈ H 1 (R), we obtain ψ (x, k) ∈ H 2 (R). Multiplying the differential equation (6.13) for ψ (x, k) with the conjugate of ψ (x, k), an integration by parts leads to 2 2 ψ (x, k) 2 dx = k 2 |ψ (x, k)|2 dx + 1 + 4α k m |ψ (x, k)|2 dx. (6.12) 4α 2 ω + 2γ R
R
R
On the other hand, the conjugate of (6.13) for ψ (x, k) yields
ψ (x, k) 2 dx = k ∗2
R
1 + 4α 2 k ∗ 4α 2 ω + 2γ
2
|ψ (x, k)|2 dx + R
Subtracting (6.13) from (6.12) yields 2 |ψ (x, k)|2 dx + 0 = k2 − k∗ R
4α 2 4α 2 ω + 2γ
m |ψ (x, k)|2 dx. (6.13) R
m |ψ (x, k)|2 dx
.
R
Now suppose that the second bracket vanishes, then 4α 2 |ψ (x, k)|2 dx = − 2 m |ψ (x, k)|2 dx, 4α ω + 2γ R
R
and combining with (6.12), we obtain 2 2 1 + 4α 2 k 2 2 ψ (x, k) 2 dx = − 4α k |ψ m k)| dx + m |ψ (x, k)|2 dx (x, 4α 2 ω + 2γ 4α 2 ω + 2γ R R R −1 1 |ψ(x, k)|2 dx. m |ψ (x, k)|2 dx = = 4α 2 ω + 2γ 4α 2 R
R
But it is a contradiction. 2 Henceforth k 2 − k ∗ =0, that is k only lies on the positive imaginary axis. To show that the zeros of a (k) are simple, we should only prove that setting k0 = iξ and a (k0 ) = 0, then da dk k=k0 = 0. By differentiating W (ϕ (x, k) , ψ (x, k)) = 2ika (k) with respect to k , and combining a (iξ ) = 0, we can see that a˙ (iξ ) =
−1 W (ϕ˙ (x, k) , ψ (x, k)) + W ϕ (x, k) , ψ˙ (x, k) . 2ξ
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Combining with ϕ (x, k) = cψ (x, k), we get
−1 1 ˙ a˙ (iξ ) = W (ϕ˙ (x, k) , ϕ (x, k)) + cW ψ (x, k) , ψ (x, k) , (6.14) 2ξ c 4α 2 k 2 +1 where k 2 = − 4α1 2 − η ω + 2αγ 2 , then η = − 4α 2 ω+2γ . So Eq. (6.3) can be reduced to ψ = −k 2 ψ −
4α 2 k 2 + 1 mψ. 4α 2 ω + 2γ
(6.15)
Differentiating (6.15) with respect to k, we obtain ψ˙ = −k 2 ψ˙ −
4α 2 k 2 + 1 8α 2 k ˙ − 2kψ − m ψ mψ. 4α 2 ω + 2γ 4α 2 ω + 2γ
Subtracting (6.15), multiplying by ψ˙ from the aboveformula, multiplying by ψ, we 2 obtain that ψ ψ˙ − ψ ψ˙ = −2kψ 2 1 + 4α 24α m . By integration on [x, ∞), it ω+2γ yields
∞
∞ 4α 2 2 ˙ ψ = −2k W ψ, ψ 2 dx + dx . mψ 4α 2 ω + 2γ x x Using a similar argument to the above, one can see that
x
x 4α 2 2 2 W (ϕ, ˙ ϕ) = −2k mϕ dx . ϕ dx + 2 −∞ −∞ 4α ω + 2γ Combining the above two forms with (6.14) and considering k purely imaginary, we have
−2kc 4α 2 2 2 a˙ (iξ ) = ψ dx + 2 mψ dx 2ξ 4α ω + 2γ R R
4α 2 |ψ|2 dx + 2 = ic m |ψ|2 dx = 0. 4α ω + 2γ R R By Lemma 6.2.1 and in terms of Theorem 2.1 in [18], we can easily get the results as follows. Theorem 6.2.2. Suppose that m ∈ H 1 (R) satisfies R (1 + |x|) |m (x)| dx < ∞ and ω + 2αγ 2 > 0, then 2 (i) The continuous spectrum of Eq. (6.3) is −∞ , 4ωα−α 2 +2γ . γ 2α2
≥ 0, there are at most finitely many eigenvalues of Eq. (6.3), all 2 lying in the interval 4ωα−α 2 +2γ , 0 . (ii) If m + ω +
With respect to the relationship between γ and the spectral parameter λ corresponding to the zeros of a (k), we have Theorem 6.2.3. γ → λ (γ ) describes a strictly decreasing smooth curve in the (γ , λ)plane.
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Proof. Now let us consider Eq. (6.1) of the isospectral eigenvalue problem. For fixed γ , we have the eigenvalue λ (γ ) and its eigenfunction ψ (x, k) accordingly. From Hurwitz’s theory (see [18]), one can see that the point(γ , λ(γ )) is a smooth curve in the (γ , λ)-plane, so we should only prove that the curve is decreasing strictly. By differentiating (6.1) with respect to γ , we get 1 α 2 − 2γ λ ψ˙ − 2 λ + λ˙ γ ψ = ψ˙ + λ (m + ω) ψ˙ + λ˙ (m + ω) ψ. 4
(6.16)
The L2 (R) inner product of (6.16) with ψ gives ˙ α 2 − 2γ λ ψ˙ , ψ − 2 λ + λγ ψ ,ψ =
1 ψ˙ , ψ + λ((m − ω) ψ˙ , ψ ) + λ˙ (m + ω) ψ, ψ
4
and the L2 (R) inner product of (6.1) with ψ˙ yields
$ # 1 α 2 − 2γ λ ψ , ψ˙ = + λ (m + ω) ψ, ψ˙ , 4
˙ ψ . Combining with (6.16), we so we have α 2 − 2γ λ ψ˙ , ψ = 41 + λ (m + ω) ψ, have ˙ −2 λ + λγ ψ , ψ = λ˙ (m + ω) ψ, ψ , after integration by parts and considering ψ (x, k) ∈ H 2 (R), one can see 2 λ + λ˙ γ ψ , ψ = λ˙ (m + ω) ψ, ψ .
(6.17)
The inner product of (6.1) with ψ, after integration by parts, leads to 1 −(α 2 − 2λγ ) ψ , ψ = ψ, ψ + λ (m + ω) ψ, ψ . 4 Combining (6.18) with (6.17), we obtain
1 λ˙ ˙ ψ , ψ = − 2λ ψ , ψ + 2λγ α 2 ψ , ψ + ψ, ψ + 2λ˙ ψ , ψ , λ 4 then λ˙ =
−2λ2 ψ ,ψ
α 2 ψ ,ψ + 41 ψ,ψ
< 0. Hence the proof of Theorem 6.2.3 is complete.
(6.18)
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6.3. The scattering data. Let ψ (t, x) be an eigenfunction corresponding to some λ in the continuous spectrum of the isospectral problem (6.3). According to the above arguments, we express ψ (t, x) as a superposition of an incident wave from x = ∞ with a reflected wave and a transmitted wave −ikx e + ρ (t, k) eikx , x → ∞, ψ (t, k, x) ≈ (6.19) , x → −∞, τ (t, k) e−ikx for some complex transmission coefficient τ (x, k) and a reflection coefficient ρ (t, k), where k ≥ 0 satisfies k 2 = − 4α1 2 − η ω + 2αγ 2 ≥ 0. τ (x, k) and ρ (t, k) define ψ (t, x) uniquely. If η is an eigenvalue of (6.3), we may express the corresponding eigenfunction ψ (t, x) as c (t, km ) e−km x , x→∞ ψm (t, km , x) ≈ m (6.20) x → −∞ ekm x , for some cm (t, km ) ∈ R, where km > 0 (independent with time t) satisfies 1 γ 2 km = + η ω + > 0. 4α 2 2α 2 On the scattering data ρ (x, k) , τ (x, k) , cm (x, km ), we obtain Theorem 6.3.1. Under the above assumption, the scattering data of the scattering problem for Eq. (1.3) are expressed as ik
ρ (t, k) = ρ (0, k) e α2
1 η +2γ
cm (km , t) = (cm (km , 0) + 1) e
t −km 2α 2
, ∀t ≥ 0;
1 η +2γ
t
τ (t, k) = τ (0, k)
, ∀t ≥ 0,
− 1 , ∀t ≥ 0.
Proof. (i) Unbounded states. The eigenfunction ψ (t, k, x) (see (6.19)) satisfies the evolution equation
1 1 1 ψt = + 2γ − u ψ u + + β ψ, (6.21) x x 2α 2 η 2 where we furthermore introduce a parameter β. By using (6.19) and the fact that u , ux → 0 as |x| → ∞, it follows that
1 1 ikx −ikx ikx ρt e = + β e−ikx + β ρ eikx . + ρ ike + 2γ −ike 2α 2 η As e−ikx and eikx are linearly independent, we obtain
ik 1 ik 1 1 1 β= + 2γ , ρt = + 2γ ρ + βρ = 2 + 2γ ρ. 2α 2 η 2α 2 η α η ik α2
1 η +2γ
Hence upon solving the above equation: ρ (t, k) = ρ (0, k) e Combining (6.21) with (6.19), as x → ∞, it implies
1 1 τt e−ikx = −ik 2 + 2γ τ e−ikx + βτ e−ikx , 2α η
t
for all t ≥ 0.
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so τt = −ik 2α1 2 η1 + 2γ τ + βτ = 0. Solving the above equation, we have τ (t, k) = τ (0, k) for all t ≥ 0. (ii) Bounded states. Considering the eigenfunction ψm (t, km , x) (see (6.20)), and combining (6.21) with (6.20), as x → ∞, it implies
1 1 cmt e−km x = + 2γ (−km ) e−km x + β cm e−km x , 2α 2 η 1 m then cmt = −k + 2γ + β cm . Considering (6.21), (6.20), as x → −∞, we have 2 η 2α 0=
km 2α 2
1 + 2γ η
ekm x + β ekm x .
km 1 Then β = − 2α + 2γ . Thus we would find that cmt = 2 η Hence upon solving the above equation, one can see that cm (km , t) = (cm (km , 0) + 1) e
−km 2α 2
1 η +2γ
−km 2α 2
t
1 η
+ 2γ (1 + cm ) .
−1
for all t > 0. Using similar arguments to the above theorem for Eq. (1.4), we obtain the results as follows. Theorem 6.3.2. Under the above assumption, the scattering data of the scattering problem for Eq. (1.4) are expressed as
ρ (t, k) = ρ (0, k) e
ik
1 1+4λω λ + 2α 2
cm (km , t) = (cm (km , 0) + 1) e Remark. Let m (x) ∈
L∗α,ω,γ
γ t
, ∀t ≥ 0; τ (t, k) = τ (0, k) , ∀t ≥ 0,
1 + 1+4λω γ t −km 2λ 4α 2
− 1 , ∀t ≥ 0.
(1 + |x|) |m (x)| dx < +∞,
= m ∈ H 3 (R) : R
−ω −
γ < m (x) ≤ 0, 2α 2
, x∈R .
The Liouville transformation is γ 41 ϕ (y) = m (x) + ω + 2 ψ (x) , 2α x % where y = −∞ m (ξ ) + ω + 2αγ 2 dξ . Convert (6.3) into the classical Sturm-Liouville problem −ϕ + Qϕ = χ ϕ, where Q (y) = and χ (:= −
1
4α 2 q(y) α2 4ωα 2 +2γ
+
qyy (y) 4q(y)
−
3qy2 (y) 16q 2 (y)
−
α2 4ωα 2 +2γ
where q (y) := m (x) + ω +
γ 2α 2
− η) are spectral parameters. According to the proof of Theorem
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6.2.3, one can see that if the initial potential satisfies m0 ∈ L∗α,ω,γ , then the Liouville substitution can be performed at any later time. Taking into account Theorem 5.3.1, and according to the IST method, we can see that solving the initial-value problem (1.3) amounts to solving a linear integrable equation and a linear second order ordinary differential equation by using similar methods as in paper [18]. This shows that (1.3) is integrable for the class L∗α,ω,γ of initial potentials. 7. Exact Peaked Solitary Wave Solutions Now we study a type of new exact peaked solitary wave solutions of the DGH Eq. (1.2). With the velocity constant c, we assume that the traveling wave solution of Eq. (1.2) is u (x, t) = v (ξ ), where ξ = x − ct. Then the above equation can be changed into the following ordinary differential equation: −cvξ + cα 2 vξ ξ ξ + 2ωvξ + 3vvξ + γ vξ ξ ξ = α 2 (2vξ vξ ξ + vvξ ξ ξ ). Integrating it once for ξ , we have 3 1 −cv + cα 2 vξ ξ + 2ωv + v 2 + γ vξ ξ = α 2 vξ2 + α 2 vvξ ξ − α 2 vξ2 + c1 . 2 2 Integrating it once again with respect to ξ , we have vξ2 =
− 21 v 3 + ( 2c − ω)v 2 + c1 v + c2 1 2 2 cα
+ 21 γ − 21 α 2 v
.
Then we have . . /
− 21 cα 2 − 21 γ + 21 α 2 v 1 3 2v
− ( 2c − ω)v 2 − c1 v − c2
Let y = − 21 cα 2 − 21 γ + 21 α 2 v, i.e., v = c1 =
1 (2y α2
dv = − |ξ | + c3 .
+ cα 2 + γ ), and
3(γ 2 + 4α 2 γ ω + 4α 4 ω2 ) γ 3 + 6α 2 γ 2 ω + 12α 4 γ ω2 + 8α 6 ω3 , c2 = , 4 −8α −16α 6
3γ 1 − 2 − 2ω , c= 2 α
then we have u (x, t) =
2 1 [ γ α2 2
+ 41 α 2 (− 3γ −2ω)]+c3 e− α |x−ct| . Hence, as 2α 2 ω+γ = α2 1
0, it has the peakon solution u (t, x) = c3 e− α |x−ct| for (1.3). 1
Acknowledgements. The authors would like to express their gratitude to Professor Walter A. Strauss for useful discussions and valuable suggestions when they visited Brown University. Research was supported by the National Nature Science Foundation of China (No: 10071033) and Nature Science Foundation of Jiangsu Province (No: BK2002003).
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References 1. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993) 2. Fuchssteiner, B., Fokas, A. S.: Symplectic structures, their Backlund transformation and hereditary symmetries. Physica D 4, 47–66 (1981) 3. Johnson, R. S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid. Mech. 457, 63–82 (2002) 4. Dullin, R., Gottwald, G., Holm, D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys.Rev. Lett. 87(9), 4501–4504 (2001) 5. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994) 6. Fisher, M., Schiff, J.: The Camassa Holm equations: conserved quantities and the initial value problem. Phy. Lett. A. 259(3), 371–376 (1999) 7. Clarkson, P.A., Mansfield, E.L., Priestley, T.J.: Symmetries of a class of nonlinear third-order partial differential equations. Math. Comput. Modelling 25 (8–9), 195–212 (1997) 8. Kraenkel, R.A., Senthilvelan, M., Zenchuk, A.I.: On the integrable perturbations of the CamassaHolm equation. J. Math. Phys. 41(5), 3160–3169 (2000) 9. Cooper, F., Shepard, H.: Solitons in the Camassa–Holm shallow water equation. Phys. Lett. A 194(4), 246–250 (1994) 10. Tian, L., Xu, G., Liu, Z.: The concave or convex peaked and smooth soliton solutions of CamassaHolm equation. Appl. Math. Mech. 123(5), 557–567 (2002) 11. Tian, L., Song, X.: New peaked solitary wave solutions of the generalized Camassa- Holm equation. Chaos, Solitons and Fractals 19(3), 621–637 (2004) 12. Tian, L., Yin, J.: New compacton solutions and solitary solutions of fully nonlinear generalized Camassa-Holm equations. Chaos, Soliton and Fractals 20(4), 289–299 (2004) 13. Constantin, A., Escher J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Sup. Pisa 26, 303–328 (1998) 14. Constantin, A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann.Inst.Fourier (Grenoble) 50, 321–362 (2000) 15. Constantin, A.: The Hamitonian structure of the Camassa-Holm equation. Exposition. Math. 15, 53–85 (1997) 16. Constantin, A., McKean, H. P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52(8), 949–982 (1999) 17. Constantin, A., Strauss, W.A.: Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002) 18. Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R.Soc . London A 457, 953–970 (2001) 19. Lenells, J.: The Scattering approach for the Camassa-Holm equation. J. Nonliear Math.Phys. 9(4), 389–393 (2002) 20. Beals, R., Sattinger, D., Szmigielski, J.: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 140, 190–206 (1998) 21. Constantin, A., Escher, J.: Wave Breaking for Nonlinear Nonlocal Shallow Water Equations. Acta Math. 181, 229–243 (1998) 22. Danchin, R.: A Few Remarks on the Camassa–Holm Equation. Differential and Integral Equations 14, 953–988 (2001) 23. Guo, BL., Liu, ZR.: Peaked wave solutions of CH − γ equation. Sci China (Ser. A) 33(4), 325–337 (2003) 24. Tang, M., Yang, C.: Extension on peaked wave solutions of CH − γ equation. Chaos, Solitons and Fractals 20, 815–825 (2004) 25. Bona, J., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans.Royal Soc. London Series A 278, 555–601 (1975) 26. Kato, T.: On the Cauchy problem for the (generalized) KdV equation. Studies in Applied Mathematics, Advances in Mathematics Supplementary. Vol.8, NewYork-London: Academic Press, 1983, pp. 93–128 27. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. J.Funct.Anal. 74, 160–197 (1987) 28. Gelfand, I.M., Dorfman, I.Ya.R.: Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl. 13, 248–262 (1979) 29. Li,Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162, 27–63 (2000) 30. Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. 46, 309–327 ( 2001)
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31. Constantin, A., Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J 47(4), 1527–1545 (1998) 32. Dunford, N., Schwartz, J.T.: Linear operators. Vol.2, New York: Wiley, 1988 33. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988) 34. Ablowitz, M., Clarkson, P.: Soliton, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press, 1993 35. Deift, P., Trubowits, E.: Inverse scattering on the line. Comm Pure Appl. Math. 32, 121–251 (1979) 36. Li, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eq. 162, 27–63 (2000) Communicated by A. Kupiainen
Commun. Math. Phys. 257, 703–723 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1355-0
Communications in
Mathematical Physics
On a Penrose Inequality with Charge Gilbert Weinstein1, , Sumio Yamada2, 1
Dept. of Mathematics, University of Alabama at Birmingham, Birmingham, AL, USA. E-mail: [email protected] 2 Mathematical Institute, Tohoku University, Sendai, Japan. E-mail: [email protected] Received: 31 May 2004 / Accepted: 8 February 2005 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfies 1 Q2 m− R+ < 0, 2 R √ where m is the total mass, R = A/4π is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality for charged black holes.
1. Introduction There has recently been much interest among geometers and mathematical relativists in inequalities bounding the total mass of initial data sets from below in terms of other geometrical quantities. The first such inequality is the Positive Mass Theorem [12, 14]. We rephrase the Riemannian version of this result as the following variational statement: among all time-symmetric asymptotically flat initial data sets for the Einstein-Vacuum Equations, flat Euclidean 3-space is the unique minimizer of the total mass. Thus, the total mass satisfies m ≥ 0 with equality if and only if the data set is isometric to R3 with the flat metric. See the next section for precise definitions. A stronger result is the Riemannian version of the Penrose Inequality, which can be stated in a similar variational vein: among all time-symmetric asymptotically flat initial data sets for the Einstein-Vacuum Equations with an outermost minimal surface of area A, the Schwarzschild slice is the unique minimizer of the total mass. In other words,
The research of the first author was supported in part by NSF Grant DMS-0205545. The research of the second author was supported in part by NSF Grant DMS-0222387.
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√ m ≥ R/2, where R = A/4π is the area radius of the outermost horizon, and equality occurs if and only if the data is isometric to the Schwarzschild slice: m 4 gij = 1 + δij . 2r When these results are phrased in this fashion, a natural question is whether similar variational characterizations of the other known stationary solutions of the Einstein Equations hold. In particular, one could ask whether among all asymptotically flat axisymmetric maximal gauge initial data sets for the Einstein-Vacuum Equations with an outermost minimal surface of area A and angular momentum J , the Kerr slice is the unique minimizer of the mass. Such a statement would imply that: 1/2 1 4J 2 2 m≥ (1) R + 2 R 2 with equality if and only if the data is isometric to the Kerr slice. Since it is not known how to define the angular momentum of a finite surface, it is necessary to assume the axisymmetry of the data set. With that hypothesis, if X is the generator of the axisymmetry, then the Komar integral: 1 J (S) = kij X i nj dA 8π S gives a quantity which depends only on the homological type of S and tends to the total angular momentum, as S tends to the sphere at infinity. A similar question can be asked with charge replacing angular momentum: is the Reissner-Nordstr¨om slice the unique minimizer of the mass among all asymptotically flat time-symmetric initial data sets for the Einstein-Maxwell Equations? This is equivalent to asking whether the following inequality holds for any data set: 1 Q2 m≥ R+ , (2) 2 R where Q is the total charge, with equality if and only if the data is a Reissner-Nordstr¨om slice. As above, the charge: 1 Q(S) = Ei ni dA 4π S depends only on the homological type of S. When the horizon is connected, inequality (2) can be proved by using the Inverse Mean Curvature flow [6, 9]. Indeed, the argument in [9] relies simply on Geroch montonicity of the Hawking mass — which still holds for the weak flow introduced by Huisken and Ilmanen in [6] — while keeping track of the scalar curvature term Rg = 2 |E|2 + |B|2 . However, when the horizon has several components the same argument yields only the following inequality: 2
min i εi Qi 1 m ≥ max Ri + , Ri 2 i where Ri and Qi are the area radii and charges of the components of the horizon i = 1, . . . , N, εi = 0 or 1, and the minimum is taken over all possible combinations. It is the purpose of this paper to point out that (2) does not hold. We prove:
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Theorem 1. There is a strongly asymptotically flat time-symmetric initial data set (M, g, E, 0) for the Einstein-Maxwell Equations such that: Q2 1 R+ m− < 0. (3) 2 R In 1984, Gibbons [4] conjectured an inequality similar to (2). However, in his conjecture, the right-hand side of (2) is taken to be additive over connected components of the horizon. Thus, Gibbons’s conjecture states that:
Q2i 1 Ri + . (4) m≥ 2 Ri i
In particular, when there is no electromagnetic field this inequality reduces to: 1 m≥ Ri , 2
(5)
i
which is stronger than the usual Riemannian Penrose inequality;
1/2 1 2 m≥ Ri . 2 i
It is not known whether (5) holds, but two Schwarzschild slices a large distance apart would seem to violate this inequality. Gibbons further conjectured that equality occurs in (4) if and only if the data is Majumdar-Papapetrou; see the next section for a description of these metrics. We note that these metrics do not actually have horizons and are not asymptotically flat in the sense of Definition 1. Instead, they have one asymptotically flat end and N asymptotically cylindrical ends which we will call necks. The cross-sections of these necks are spheres with mean curvature tending to zero as the surfaces go further down the end. Our construction is based on the fact that the Majumdar-Papapetrou metrics ‘violate’ (2), say with N = 2, and m1 = m2 . They do not strictly speaking violate (2) since they are not asymptotically flat and do not possess horizons. In order to remedy these failures, we glue two such copies along the necks. The gluing procedure we use is an adaptation of the conformal perturbation method developed for the vacuum case in [7]. In fact in our setting, some of the technical difficulties arising from the generality of the construction in [7] are absent. However, while it is easy to show the existence of a two-component minimal surface in the resulting metric, we must also show that (2) is violated with R the area radius of the outermost horizon. This requires ruling out minimal surfaces outside the necks which we can accomplish by letting m → 0 which is equivalent after rescaling to taking the two masses in the initial Majumdar-Papapetrou far apart. We point out that this counter-example has little to do with the Cosmic Censorship conjecture. In fact, as pointed out by Jang [9], inequality (2) is equivalent to: m − m2 − Q 2 ≤ R ≤ m + m2 − Q 2 , and only the upper bound would follow from Cosmic Censorship using Penrose’s heuristic argument. Our counter-example violates the lower bound. The paper is organized as follows. In the next section, we define some terms, and set-up the notation. In Sect. 3, we carry out the gluing. In the last section, we show that the parameters can be chosen so that the resulting initial data violates (2).
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2. Preliminaries Definition 1. Let (M, g) be a 3-dimensional Riemannian manifold. We say that (M, g) is strongly asymptotically flat (SAF) if there is a compact set K ⊂ M such that M \ K is the disjoint union of finitely many ends Nν , ν = 1, . . . , k, each end Nν is diffeomorphic to R3 minus a ball and admits a coordinate system in which the metric satisfies: 2,α gij − δij ∈ C−1 (Nν ).
2,α Here C−1 (N ν ) denotes the class of functions φ such that r |φ|, r 2 |∂φ|, r 3 ∂ 2 φ and r 3+α |x − y|−α ∂ 2 φ(x) − ∂ 2 φ(y) are bounded. While the bound is coordinate dependent, the set of function is independent of coordinates. We will focus our attention on one end which we will denote by N+ . We will denote all the other ends collectively as N− . In fact, in this paper we are dealing exclusively with two-ended SAF manifolds so that N− consists of only one end. By adding a point ∞− (or in the general case k − 1 points) at infinity in N− and conformally compactifying, we obtain an asymptotically flat Riemannian manifold with one end. We now consider the class S of smooth surfaces S which bound a compact region such that ∞− ∈ . In this class, it makes sense to speak of the outer unit normal. If S1 , S2 ∈ S, we will say that S1 encloses S2 if the corresponding regions 1 and 2 satisfy 1 ⊃ 2 . If (M, g) is strongly asymptotically flat, the total mass m of the end N+ is defined by: 1 gij,j − gjj,i ni dA, m= lim 16π r→∞ Sr where Sr is the Euclidean coordinate sphere in N+ , n its unit normal in δ, and dA the area element induced on Sr from δ. Definition 2. A horizon S is a minimal surface in (M, g) which belongs to S. An outermost horizon is a horizon which is not enclosed within any other horizon. A surface S ∈ S is outer minimizing if it has area no greater than any other surface which encloses it. Note that for r large enough, Sr ∈ S and has positive mean curvature with respect to its outer unit normal. Thus, by minimizing area over all surfaces in S which enclose the outermost horizon S, and are enclosed in Sr , we obtain a minimal surface S1 which encloses S. It then follows from the outermost property of S that S = S1 ; see [11, Theorem 1’, p. 645]. We conclude that an outermost horizon is also outer minimizing, a fact which will be used in the last section. A time-symmetric initial data set (M, g, E, B) for the Einstein-Maxwell Equations consists of a Riemannian manifold (M, g), and two vector fields E and B on M such that: Rg = 2 |E|2g + |B|2g , divg E = divg B = 0, E × B = 0, g(B, ng ) dA = 0, S
where Rg is the scalar curvature of g, and S ⊂ M is an arbitrary closed surface with normal ng of unit length in g. We say that the set (M, g, E, B) is strongly asymptotically 2,α 2,α , B ∈ C−3 . flat if (M, g) is SAF, and if E ∈ C−2
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Choose N > 0, mk > 0, and pk ∈ R3 for k = 1, . . . , N, and let rk denote the Euclidean distance to pk in R3 . The Majumdar-Papapetrou solutions are given by: u= 1+
N mk k=1
rk
1/2 ,
gij = u4 δij ,
Ei = 2∇i log u,
Bi = 0.
(6)
When N = 1, this is simply the extreme case m = |Q| of the Reissner-Nordstr¨om data set. Note that if we take E− = −2∇ log u instead of E = 2∇ log u, we get another solution with charges of opposite sign. For simplicity, we will restrict ourselves to the case N = 2, m1 = m2 = m, i.e. 1/2 . It is not difficult to check that (M, g, E, 0) satisfies the u = 1 + m/r1 + m/r2 Einstein-Maxwell time-symmetric constraints. In fact, the metric −u−4 dt 2 +g is a static solution of the Einstein-Maxwell equations. Let r denote the Euclidean distance from the origin. We denote by Bi (ρ) = {ri < ρ} the Euclidean ball of radius ρ centered at pi , and by B0 (ρ) = {r < ρ} the Euclidean ball of radius ρ centered at the origin. Note that for R large enough N = R3 \ B0 (R) equipped with the metric g is a SAF end, and the necks Bi (ρ) \ {pi } are asymptotically cylindrical. It is easy to check that the total mass µ of N is 2m, the total charge Q = S g(E, n) dA is 2m, while the total cross sectional √ area A of both necks is asymptotically 8πm2 , i.e., R = 2m. Thus, we get: √ Q2 3 1 √ 1 R+ = 2m − ( 2m + 2 2m) = m 2 − √ µ− < 0. 2 R 2 2 However (M, g) admits no horizon. In the next section, we remedy this by gluing at the necks a second copy of opposite charges. The solution of the constraints is achieved through a conformal perturbation argument. We will then show in Sect. 4 that the resulting data set possesses a horizon which violates (2). 3. The Gluing Let (M± , g± , E± , 0) be two copies of the Majumdar-Papapetrou data, with E− = −E+ . In this section, we show that we can glue these two copies along their necks. This gluing will be performed by a perturbation method with perturbation parameter T > 0 large. Whenever a possible ambiguity might arise, we use a subscript (or superscript) + (or − respectively) to indicate a quantity associated with M+ (or M− respectively). For convenience, we take p1 = (0, 0, 1) and p2 = (0, 0, −1). The gluing is accomplished in three steps. In the first step, we truncate the necks at ri = e−T , and introduce cut-offs in the regions e−T +1 < ri± < e−T +2 to obtain a transition to round cylinders. This yields data on M+ and M− which matches in the regions e−T < ri± < e−T +1 of the necks. We can then identify the corresponding boundaries ri± = e−T in M+ and M− ˆ g, ˆ 0). However, this data no longer satisfies creating a two-ended SAF data set (M, ˆ E, the constraint equations in the cut-off regions. In the second step, we restore the diverˆ gence constraint divgˆ Eˆ = 0, Eˆ = Eˆ −∇ϕ, by solving a linear problem gˆ ϕ = divgˆ E, ϕ → 0 at ∞. Finally, in the last step, we use a perturbation argument to find a conˆ φ 4 g, g, 0) = (M, formal deformation (M, ˜ E, ˆ φ −6 Eˆ ) which satisfies the constraints. It is easy to see that the divergence constraint is automatically preserved under the above = 0. The Gauss conformal transformation g → φ 4 g, Eˆ → φ −6 Eˆ , i.e., we have divg˜ E
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2 is then satisfied if and only if φ satisfies the following nonlinear constraint Rg˜ = |E| g˜ equation [8]: Lgˆ φ = −
|Eˆ |2gˆ 4φ 3
(7)
,
where Lgˆ = gˆ − 18 Rgˆ is the conformal Laplacian of g. ˆ Section 3.3 is therefore devoted to showing that for T large enough, there is a positive solution φ of (7) such that φ − 1 2,α is small in C−1 . This gluing technique is an adaptation of [7]. 3.0. Function spaces and elliptic theory. Let (M, g) be a SAF manifold with K ⊂ M compact and M \ K the disjoint union of finitely many ends Nν . Let σ ≥ 1 be a weight function on M such that σ = 1 on K, and equals the Euclidean distance r on each end k,α Nν for r large enough. Let C−β (M) be the set of functions φ on M whose k th order derivatives are H¨older continuous and for which the norm φ C k,α defined below is −β
finite: φ C k = −β
[D k φ]α,−β =
k β+i i D φ σ
C0
i=0
sup
,
σ (x, y)β+α
x k
Py D φ(y) − D k φ(x)
0
dist(x, y)α
,
α φ C k,α = φ C k + [D k φ]C−β−k . −β
−β
Here D i φ represents the tensor of i th order derivatives of φ, ρ is the injectivity radius of (M, g), σ (x, y) = max{σ (x), σ (y)}, and Pyx is parallel translation along the shortest geodesic from y to x. Theorem 2. Let (M, g) be a SAF manifold. 0 (M) and φ ∈ C 0,α (M), then φ ∈ C 2,α (M) and (a) Let φ ∈ C−β g −β −β−2
φ C 2,α ≤ C( φ C 0 + g φ C 0,α ). −β
−β
−β−2
(8)
0,α 2,α (b) Let 0 < β < 1, ν > 2, and let h ∈ C−ν (M). If the operator g − h : C−β (M) → 0,α (M) is injective then it is an isomorphism. C−β−2
This theorem is stated in [10], but the reader is referred to [3] for the proof. Unfortunately, the proof of part (b) in [3] has a small gap, which is nevertheless easily remedied. For details, please refer to [13, Appendix], where a complete proof is given for the case M = R3 . The proof for general SAF manifolds is a straightforward combination of the arguments in [3] and [13].
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3.1. Preparation. Fix T > 0, and let χ (r) be a smooth positive nondecreasing cutoff function such that χ (r) = 1 for r > e−T +2 , and χ (r) = 0 for r < e−T +1 . Let χi = χ (ri ), i = 1, 2, and define: m m 1/2 uˆ = χ1 χ2 + χ2 + χ1 , r1 r2 We note that:
ˆ 4 ≤ Ce−T ,
1 − (u/u)
gˆ = uˆ 4 δ =
4 uˆ g, u
∇ log uˆ − ∇ log u ≤ Ce−T , g
where C is a constant independent of T . This implies
gˆ − g ≤ Ce−T , ˆ 2
≤ Ce−T , − 2| E|
R g ˆ gˆ g
Eˆ i = 2∇i log u. ˆ
g (u/u) ˆ ≤ Ce−T , (9)
divgˆ Eˆ ≤ Ce−T .
(10)
Introduce the notations: Bi (ρ) = {ri < ρ}, D(ρ) = B1 (ρ) ∪ B2 (ρ), −T i (ρ) = {e ≤ ri < ρ}, (ρ) = 1 (ρ) ∪ 2 (ρ).
(11) (12)
On i (e−T +1 ), gˆ = m2 (dri2 /ri2 + dω2 ) is a round cylindrical metric with dω2 the standard metric on the unit sphere, and Eˆ = dri /ri is parallel. Thus, if we take two copies M± = R3 \ D(e−T ), then both the metrics gˆ ± = gˆ and the vector fields Eˆ ± = ±Eˆ match on ri± = e−T , and we can identify these boundaries to form a doubly-connectedˆ sum Mˆ = M+ #M− . We will denote the metric on Mˆ by gˆ and the vector field by E. ˆ g) We note that (M, ˆ is a two-ended SAF manifold. We denote ˆ i (ρ) = i+ (ρ) ∪ i− (ρ) ˆ and (ρ) = + (ρ) ∪ − (ρ). We have suppressed the dependence on T in order not to encumber the notation. We now fix the weight function σ = σ (r) to be 1 on {r ± ≤ 3} in M± , monotone in r, and equal to r ± on {r ± > 4}. In addition, we can assume that it is even with respect to reflections across the cuts ∂M+ = {r1 = e−T } ∪ {r2 = r −T }. Note that since u/u ˆ =1 ˆ −T +2 ), the quantities in (10) vanish outside this set, hence these estimates outside (e hold also with any weighted norms. In particular: −T −T ˆ ˆ 2 E ≤ Ce , . (13) Rgˆ − 2|E| div gˆ 0,α ≤ Ce 0,α gˆ C−3
C−3
Throughout the rest of this section, C, C , c will denote various constants independent of T . In order to simplify the notation, we may at times change the value of such constants. This abuse of notation can be justified by simply taking the maximum of the previous and current value of the constant. We will need the following essentially local elliptic estimate. 0,α Proposition 1. Let 0 < β < 1, ν > 2, and let h ∈ C−ν satisfy h ≥ 0. There is a constant C independent of T , such that for each T large enough, φ C 2,α ≤ C φ C 0 + ( gˆ − h)φ C 0,α . (14) −β
−β
−β−2
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Proof. On either end ± = M± \ B0± (4), we can use an argument using local estimates and the scaling of annuli as in [13, Prop. 26] to get a weighted estimate: φ C 2,α ( ) ≤ C φ C 0 ( ) + ( gˆ − h)φ C 0,α ( ) −β
±
−β
±
−β−2
±
with a constant C independent of T , where ± = M± \B0± (3). Now, let K = Mˆ \ {r + ≥ gˆ 5} ∪ {r − ≥ 5} , then K can be covered by finitely many geodesic balls Bqi (ρ) of radius ρ > 0 sufficiently small, so that the elliptic constant of gˆ written in normal coordinates gˆ on Bqi (2ρ) is uniformly bounded above and below. While the number of balls depends on T , ρ can be chosen independently of T . We have local elliptic estimates: φ C 2,α (B gˆ (ρ)) ≤ C φ C 0 (B gˆ (2ρ)) + ( gˆ − h)φ C 0,α (B gˆ (2ρ)) , qi
qi
qi
where C depends on ρ but is independent of i or T . Collecting these estimates yields (14). 3.2. The divergence constraint. In this section, we restore the divergence constraint by solving the following linear problem: gˆ ϕ = f,
ϕ → 0 at ∞,
ˆ We must also ensure that ϕ tends to zero when T tends to infinity. where f = divgˆ (E). Proposition 2. For each m > 0 small enough, and each T large enough, there is a 2,α unique solution ϕ ∈ C−1 of the equation: gˆ ϕ = f, ˆ where f = divgˆ (E). ˆ Furthermore, on M, ϕ C 2,α ≤ CT 2 e−T , −1
(15)
where the constant C is independent of T . 2,α Proof. The existence of a solution ϕ ∈ C−1 is standard, see e.g. [3]. The smallness of ϕ, inequality (15), will follow from the elliptic estimates in Proposition 1 once we obtain a weighted supremum bound:
sup σ |ϕ| ≤ CT 2 e−T ,
(16)
Mˆ
where C is independent of T . This is obtained by a comparison argument using the ˆ −T +2 ), and P = maximum Note that the function f has supp f ⊂ (e T principle. supT e sup |f | < ∞. Furthermore, f is odd with respect to reflection across the cuts ∂M+ which implies that ϕ is also odd, hence ϕ = 0 on ∂M+ . Now let: r log(s) −T ψ(r) = −e ds. −T s(s + m) e We claim that if m is small enough and T is large enough, then w = ψ(r1 ) + ψ(r2 ) has the following properties on M+ :
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(i) 0 < w ≤ m−1 T 2 e−T on M+ . (ii) gˆ w ≤ 0 on M+ . (iii) gˆ w ≤ −ce−T on (e−T +2 ) for some c > 0 independent of T . These properties imply that ϕ−c−1 P w satisfies gˆ (ϕ−c−1 P w) ≥ 0, and ϕ−c−1 P w ≤ 0 both on ∂M+ and at ∞. Thus, we get ϕ ≤ c−1 P w on M+ . Similarly, by considering the function ϕ + c−1 P w, we get ϕ ≥ −c−1 P w on M+ . This yields an unweighted supremum estimate: sup |ϕ| ≤ M+
P 2 −T T e . cm
(17)
By symmetry, the same estimate holds on M− . Now, in order to get the weighted estimate (16), let = M+ \ B0+ (3), and let v be the solution of the following problem: g v = 0 in ,
v = 1 on ∂ ,
v → 0 at ∞.
There is a constant C such that 0 < v ≤ Cσ −1 . Let K = P /cm, then the functions ±ϕ + KT 2 e−T v are harmonic in with respect to g = g, ˆ are non-negative on ∂ , and tends to 0 at ∞, hence by the maximum principle ±ϕ + KT 2 e−T v ≥ 0 in . Hence, we obtain σ |ϕ| ≤ KCT 2 e−T on . Combining with (17), the weighted estimate (16) follows. It remains to prove the claims (i)–(iii). Denote ψi = ψ(ri ), and note that 1 1 2 −T log s −T max ψi = ψ(1) = −e ds ≤ T e , 2m e−T s(s + m) whence w ≤ m−1 T 2 e−T . A similar estimate shows that ψ(1) > T 2 e−T /4m. On the other hand, ∞ log s −T ψ(1) − ψ(∞) = e ds ≤ e−T < ψ(1), s(s + m) 1 provided T is large enough. We conclude that w > 0 if T is large enough proving (i). In order to establish (ii) and (iii), we first note that it is sufficient to prove these with gˆ replaced by g. Indeed, suppose that (i) and (ii) hold with g instead of g. ˆ Then we have: u 4 u 2 g w + g ∇(u/u) ˆ 2 , ∇w gˆ w = uˆ uˆ and u/u ˆ = 1 outside (e−T +2 ), while on (e−T +2 ):
C
|∇w|g ≤ 2 T e−T , ˆ 2 ≤ Ce−T .
∇(u/u) g m It follows that (i) and (ii) also hold with gˆ once we replace c by say c/2, provided T is large enough. We now turn to proving (i) and (ii) with respect to g. Let g1 = u41 δ be the one-blackhole Majumdar-Papapetrou metric, i.e., u21 = 1 + m/r1 . One easily calculates: g1 ψ1 = −
e−T . (r1 + m)3
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Let θ = ∇r1 · ∇r2 denote the inner product of ∇r1 and ∇r2 with respect to δ, then: g ψ1 = u−6 divδ (u2 ∇ψ1 ) m = u−6 divδ u21 + ∇ψ1 r2
u 6 1 2 m 1 = g1 ψ1 + 6 divδ u1 ∇ψ1 u u r2 u21
u 6 mu21 m e−T 1 1 =− 1+ · ∇ψ1 + 6 ∇ u u r2 u21 (r1 + m)3 r2 u21
u 6 mr12 log r1 m m2 log r1 e−T 1 1+ =− + − . θ u (r1 + m)3 r2 u21 r2 (r1 + m)u21 r22 (r1 + m) Note that r1 > 1 on B2 (1) hence, there, we can estimate g ψ1 above by the only positive term on the right-hand side:
e−T |log r1 | me−T r2 log r1
≤ g ψ1 ≤ r2 . θ (r2 + mr2 /r1 + m)3 r1 (r1 + m) m2 r12 Furthermore, as r1 → 0, then g ψ1 → −e−T /m3 . It follows that, provided m < 1, one can choose ε > 0 independent of T and m such that:
e−T /4m3 , when r2 < ε −e−T /2m3 , when r1 < ε.
g ψ1 ≤
Now, we can choose m > 0 small enough, so that gˆ ψ1 ≤ 0 when r1 , r2 ≥ ε. By symmetry, we have analogous estimates for g ψ2 . We conclude that: g w = g ψ1 + g ψ2 ≤ 0, g w ≤ −
e−T 4m3
,
when r1 , r2 ≥ ε,
on (ε).
Properties (ii) and (iii) now follow provided T > − log ε + 2. This completes the proof of Proposition 2. Defining Eˆ = Eˆ − ∇ϕ, we now have divgˆ (Eˆ ) = 0, and in view of (15) and (13): Rgˆ − 2|Eˆ |2gˆ
0,α C−3
≤ CT 2 e−T .
(18)
3.3. The Gauss constraint. In this section, we prove that for each T large enough, there 2,α is a positive solution φ ∈ 1 + C−1 (M) of Eq. (7). We first prove the following estimate which gives a uniform bound on the inverse of the linearized operator associated with (7). The proof is adapted from [7, Prop. 8].
On a Penrose Inequality with Charge
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Proposition 3. Let h=
1 3 Rgˆ + |Eˆ |2gˆ . 8 4
2,α , then Then there is a constant C independent of T such that if φ ∈ C−2/3 φ C 2,α ≤ C ( gˆ − h)φ C 0,α . −2/3
−8/3
Remark 1. For T bounded, this follows from Theorem 2. 2,α Proof. Suppose the contrary. Then there is a sequence Tj → ∞, and φj ∈ C−2/3 satisfying φj 2,α = 1 ∀j, ( gˆ − h)φj 0,α → 0 as j → ∞. (19) C C −2/3
−8/3
By Proposition 1, we have φj 2,α ≤ C φj 0 C C
−2/3
−2/3
+ ( gˆ − h)φj C 0,α , −8/3
with C independent of j . Hence, in view of (19), we have ε > 0 such that ε ≤ φj 0 ≤ 1, C−2/3
(20)
for all j . We now consider the following two cases: (i) There is τ > 0 such that for any δ > 0, we have lim sup φj C 0 ((δ)) ≥ τ. ˆ j
(ii) For every τ > 0, there exists δ > 0 so that lim sup φj C 0 ((δ)) < τ. ˆ j
ˆ Note that (δ) is the union of the two necks cut at ri± = δ. Case (i). For each integer k large enough, take δk = e−k in (i). Then there
is jk large ˆ k ) with φ(pjk ) ≥ τ/2. Withenough so that Tjk > k and so that there exists pk ∈ (δ out loss of generality we may assume that pjk ∈ 1+ (δk ). Furthermore, by passing to a subsequence, we may assume that jk = k, i.e., Tk > k, and pk ∈ 1+ (δk ). We define a coordinate s on Mˆ by s = ±(log r1 + T ) on M± . Denote sk = s(pk ), then it follows that 0 ≤ sk < Tk − k. Now let k = {sk − k/2 < s < sk + k/2}. The part {s = sk + k/2} of the boundary of k has r1+ coordinate equal to: exp(sk + k/2 − Tk ) < e−k/2 → 0,
as k → ∞.
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G. Weinstein, S. Yamada
A similar estimate holds for the other part of the boundary. It follows that the metric gˆ on k converges to m2 (ds 2 + dω2 ) as k → ∞. Let (, g0 ) denote the standard round cylinder with the metric g0 = m2 (ds 2 + dω2 ). We will identify the points on k with those of via the identity map induced by the (s, ω) coordinates. Observe
that ∪k = . Using the compactness of the embedding C 2,α (k ) → C 2,α (k ), 0 ≤ α < α, we can now select a subsequence, which we now denote φk again, and a
function φ0 on such that φk → φ0 in C 2,α (k ) for each fixed k. Furthermore, there is a point p0 in the cross-section {s = 0} of such that |φ0 (p0 )| ≥ τ/2, hence φ0 is not identically zero. The scalar curvature Rgˆ on k converges to 2/m2 and |Eˆ |2gˆ converges to 1/m2 . Thus, the coefficients of T = gˆ − h converge uniformly on compact sets to the coefficients of 1 T0 = g0 − 2 . m
Hence we get Tφk → T0 φ0 in C 0 (k ). Since we also have Tφk → 0 in C 0,α (k ) by (19), we conclude that φ0 satisfies the linear equation 1 φ0 = 0 m2 on . Since φ0 is nontrivial, it has exponential growth in s either as s → ∞ or as s → −∞, in contradiction to (20). Case (ii). We begin the treatment of this case with the following lemma: g0 φ0 −
Lemma 1. Suppose φj satisfies (19) and (ii), and let A+ δ ⊂ M+ be the twice perforated ball Aδ = B0 (3) \ D(δ/2). Then for each δ > 0, there holds φj C 1,α (A ) → 0 as δ j → ∞. Proof. Suppose not, and let jk be a subsequence such that φjk converges to φ0 in
C 1,α (Aδ ), α < α. Then φ0 is not identically zero on Aδ , hence since h > 0 on Aδ , we have: lim hφj2k = hφ02 > 0. k
Aδ
We now proceed to show that
Aδ
lim sup j
Aδ
hφj2 = 0,
(21)
leading to a contradiction. Without loss of generality, we may assume, by passing to a subsequence, that hφj2 → lim sup hφj2 . Aδ
j
Aδ
If χ is any smooth cut-off function of compact support in M+ , with 0 ≤ χ ≤ 1, and χ = 1 in Aδ , then
2 2 hφj ≤ χ 2 ∇φj gˆ + hφj2 Aδ M+ 2 =− χ φj ( gˆ − h)φj − 2 χ φj g(∇χ ˆ , ∇φj ). M+
M+
On a Penrose Inequality with Charge
715
We will now choose cut-offs χk , and a subsequence jk along which both of the terms on the right-hand side tend
to zero, proving (21). By (ii), for each integer k we can choose ˆ k ) for all j large enough. Now, we can choose 0 < δk < δ such that φj < 1/k on (δ χk supported on B0 (k) \ D(δk /2), with supp ∇χk ⊂ B0 (k) \ B0 (3) ∪ D(δk ) \ D(δk /2) , and satisfying: |∇χk |gˆ ≤ C/k, |∇χk |gˆ ≤ C,
on B0 (k) \ B0 (3), on D(δk ) \ D(δk /2).
Finally, by (19), we can choose jk > jk−1 , so that 1 −10/3 ( gˆ − h)φjk C 0,α ≤ . σ −8/3 3 k R \B0 (3) It then follows that
−
χk2 φjk ( gˆ − h)φjk <
M+
and
− M+
χk φjk g(∇χ ˆ k , ∇φjk ) ≤ C D(δk )\D(δk /2)
1 , k
φj + Ck −1 k
σ −7/3
B0 (k)\B0 (3) k
dr ≤ Ck −1 + Ck −1 1/3 r 3 ≤ C k −1 + k −1/3 . This completes the proof of Lemma 1.
Now choose δ > 0 so that lim sup φj C 0 ((δ)) < ε, ˆ
(22)
j
where ε is defined by (20), and define a new manifold (M∗ , g∗ ) diffeomorphic to R3 by extending smoothly the metric gˆ on R3 \ D(δ/2) over D(δ/2). Then extend smoothly to D(δ/2) also the potential function h so that the extended potential h∗ satisfies h∗ ≥ 0 on R3 . Let χ be a smooth cut-off function on R3 with 0 ≤ χ ≤ 1, χ = 1 outside D(δ), and χ = 0 on D(δ/2). Taking the values of φj from M+ , we can view χ φj as a function on M∗ , and we find: ( g∗ − h∗ )χ φj = ( gˆ − h)χ φj = χ ( − h)φj + 2g(∇χ ˆ , ∇φj ) + φj gˆ χ . Hence, we can estimate: ( g − h∗ )χ φj 0,α ∗ C
−8/3 (M∗ )
≤ ( − h)φj C 0,α + C φj C 1,α (A ) → 0, −8/3
δ
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G. Weinstein, S. Yamada
by (19) and Lemma 1. It then follows by Theorem 2 part (b) applied to (M∗ , g∗ ) that χ φj 2,α → 0. C (M ) ∗
−2/3
Thus, we obtain
φj
→ 0.
φj
→ 0,
0 C−2/3 (M+ \ + (δ))
Similarly, we obtain
0 C−2/3 (M− \ − (δ))
and it follows that
φj
→ 0.
0 ˆ (δ)) ˆ C−2/3 (M\
Combining with (22), we conclude that lim sup φj C 0
ˆ
−2/3 (M)
j
< ε,
in contradiction to (20). This completes the proof of Proposition 3.
We can now prove the main result of this section. Proposition 4. For each m > 0 small enough and for each T large enough there is a 2,α solution φ ∈ 1 + C−1 (M) of Lgˆ φ = −
|Eˆ |2gˆ 4φ 3
(23)
.
Furthermore, as T → ∞, this solution satisfies φ − 1 C 2,α → 0. −1
2,α 0,α → C−8/3 be the following nonlinear operator: Proof. Let N : 1 + C−2/3
N(1 + ψ) = Lgˆ (1 + ψ) +
|Eˆ |2gˆ 4(1 + ψ)3
.
(24)
The linearization of N about 1 is 3 2,α 0,α dN = Lgˆ − |Eˆ |2gˆ : C−2/3 → C−8/3 , 4 0,α 2,α and according to Proposition 3, dN−1 : C−8/3 → C−2/3 is uniformly bounded, i.e., there is a constant C independent of T such that −1 dN ψ 2,α ≤ C ψ C 0,α . C−2/3
−8/3
Now consider the ‘quadratic part’ of N: Q(ψ) = N(1 + ψ) − N(1) − dN(ψ).
On a Penrose Inequality with Charge
717
We have Q(ψ) =
|Eˆ |2gˆ (6 + 8ψ + 3ψ 2 ) 4(1 + ψ)3
ψ 2,
hence it follows that there is a constant C independent of T such that if η > 0 is sufficiently small, and ψ C 2,α < η, then the following holds: −2/3
Q(ψ) C 0,α ≤ Cη2 ,
(25)
−8/3
Q(ψ1 ) − Q(ψ2 ) C 0,α ≤ 2Cη ψ1 − ψ2 C 2,α . −8/3
(26)
−2/3
Now, choose 0 < λ < 1, η > 0 such that η < λ/2C 2 , and T > 0 such that T 2 e−T < η2 . 2,α Then, if B is the ball of radius η in C−2/3 , the map F given by F(ψ) = −dN−1 N(1) + Q(ψ) maps B into B and is a contraction. Indeed, in view of (18) and (25), we have F(ψ) C 2,α ≤ C N(1) C 0,α + Q(ψ) C 0,α ≤ C 2 (T 2 e−T + η2 ) < η, −2/3
−8/3
−8/3
and in view of (26), F(ψ1 ) − F(ψ2 ) C 2,α ≤ C Q(ψ1 ) − Q(ψ2 ) C 0,α −2/3
−8/3
≤ 2C η ψ1 − ψ2 C 2,α < λ ψ1 − ψ2 C 2,α . 2
−2/3
−2/3
It follows that F has a fixed point ψ in B which satisfies N(1 + ψ) = N(1) + dN(ψ) + Q(ψ) = 0. Furthermore, note that if T → ∞, one can choose η → 0. Thus we have ψ C 2,α → 0. −2/3
It also follows from (24) that gˆ ψ =
|Eˆ |2gˆ 1 1 Rgˆ ψ + − Rgˆ . 8 8 4(1 + ψ)3
0,α We will now show that the right-hand side above tends to zero as T → ∞ in C−3 ∩ L1 . Indeed, we have
|Eˆ |2gˆ 1 1 − Rgˆ Rgˆ ψ + 3 4(1 + ψ) 8 8 |Eˆ |2gˆ 1 1 3 Rgˆ − 2|Eˆ |2gˆ − = Rgˆ ψ − − 1 (1 + ψ) 8 8 4(1 + ψ 3 ) |Eˆ |2gˆ (3 + 3ψ + ψ 2 ) 1 1 ˆ + ∇ϕ, ∇ϕ) + ˆ 2 − 1 g(2 Rgˆ − 2|E| ˆ E ψ. = Rgˆ ψ − gˆ 8 8 4 4(1 + ψ 3 )
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G. Weinstein, S. Yamada
0,α We now proceed to check that each of the terms above tends to zero in C−3 ∩ L1 as T → ∞. The second term above tends to zero by (13), and the fact that it is supported on a set of uniformly bounded volume. For the other three terms, we use the fact that if 0,α fi ∈ C−β , i = 1, 2, and β1 + β2 > 3, then i
f1 f2 C 0,α ∩L1 = f1 f2 C 0,α + f1 f2 L1 ≤ C f1 C 0,α f2 C 0,α . −3
−β1
−3
−β2
If one of the factors on the right-hand side of the inequality tends to zero and the other is bounded, then the left-hand side of the inequality tends to zero. The first and last term above are of the form f ψ, with f C 0,α bounded and ψ C 0,α → 0. The third term is −4
−2/3
of the form f |∇ϕ|gˆ with f C 0,α bounded and ∇ϕ C 0,α → 0. We conclude that −2
−2
gˆ ψ
0,α C−3 ∩L1
→ 0.
(27)
The result will now follow from the following lemma. 0,α 0,α Lemma 2. Suppose that ψ ∈ C−2/3 and gˆ ψ ∈ C−3 ∩ L1 . Then there is a constant C independent of T such that ψ 2,α ≤ C gˆ ψ 0,α 1 + ψ 2,α . C−3 ∩L
C−1
C−2/3
Proof of Lemma 2. The proof of this lemma is based on the proof of Proposition 29 0,α in [13, Appendix]. There, it is proved that if v is a function on R3 with v ∈ C−2/3 , and
0,α ∩ L1 , then gˆ v ∈ C−3
v C 2,α ≤ C gˆ v C 0,α ∩L1 . −1
(28)
−3
Let R = {r > R} ⊂ M+ . Clearly, for any finite R ≥ 3, the two norms · C 2,α (M \ ) + R −1 and · C 2,α (M \ ) are equivalent. Let χ be a smooth cut-off function with 0 ≤ χ ≤ 1, −2/3
+
R
χ = 0 on M \ 3 and χ = 1 on 4 . Then v = χ ψ can be viewed as a function on R3 , 2,α and v ∈ C−2/3 . We have ˆ , ∇ψ) + ψ gˆ χ . gˆ v = χ gˆ ψ + 2g(∇χ The last two terms above are supported on the annulus 3 \ 4 , hence we can estimate 2g(∇χ ˆ , ∇ψ) + ψ gˆ χ C 0,α ∩L1 ≤ C ψ C 2,α , −3
−2/3
while for the first term we clearly have χ gˆ ψ 0,α 1 ≤ C gˆ ψ 0,α 1 . C ∩L C ∩L −3
−3
Thus, we obtain gˆ v
0,α C−3 ∩L1
≤ C gˆ ψ C 0,α ∩L1 + ψ C 2,α . −3
−2/3
On a Penrose Inequality with Charge
719
We now conclude from (28) that ψ C 2,α ( ) ≤ C gˆ v C 0,α ∩L1 ≤ C gˆ ψ C 0,α ∩L1 + ψ C 2,α . −1
−3
4
−3
−2/3
Hence, we have ψ C 2,α (M −1
+)
≤ C ψ C 2,α ( ) + ψ C 2,α (M \ ) + 4 5 −1 −1 ≤ C gˆ ψ 0,α 1 + ψ 2,α . C−3 ∩L
C−2/3
A similar estimate holds on M− . This completes the proof of Lemma 2.
Taking φ = 1 + ψ, we see that φ satisfies Eq. (23), and by (27) and Lemma 2 we have φ − 1 C 2,α → 0 as T → ∞. This completes the proof of Proposition 4. −1
We have shown that for each m > 0 sufficiently small and for each T sufficiently g, 0) of the Einstein-Maxwell constraints, large, there is a two-ended solution (M, ˜ E, 4 4 4 ˆ ˜ ˜ with M = M, g˜ = φ uˆ δ = φ g, where φ = φ u/u ˆ and g is the Majumdar-Papapetrou solution. We note that for any η > 0, we can assure that φ˜ − 1 C 2,α < η, −1
1,α < η E − E C −2
(29)
by taking T large enough. For the sake of simplicity, we now rename φ˜ to be φ. Furthermore, we note that this solution admits an involutive, charge-reversing, symmetry with fixed-point set 0 . 4. The Outermost Horizon In this section, we show that with m > 0 fixed and sufficiently small, we can adjust the perturbation parameter η > 0 to be small enough so that the area A˜ of the outermost horizon in the conformal perturbation g˜ is no greater than 8π λ2 m2 , where λ − 1 > 0 √ is arbitrarily small, i.e., R ≤ 2λm. Furthermore, if the perturbation parameter η > 0 is small enough, we can assure that the total mass m of g˜ is no greater than 2λm of E satisfies Q ≥ Q/λ, where Q = 2m. Now, the function and that the charge Q fQ (x) = x + Q2 /λ2 x is non-increasing for 0 < x < Q/λ. Thus, if we choose λ so that 1<λ<
√
2 − 1/2
−1/4
< 21/4 ,
we get: 1 m − R+ 2
√
2 2 Q 1 Q2 1 R+ 2 ≤m − ≤ λm 2 − √ − 4 < 0, 2 λ λ R R 2
This proves Theorem 1. We will use the following elementary lemma.
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G. Weinstein, S. Yamada
Lemma 3. Let M be a Riemannian manifold, and let S ⊂ M be a compact hypersurface with unit normal n. Let v be a smooth function on M, and suppose that the maximum of v over S is taken at a point q ∈ / ∂S, where ∇v = 0. Then
v − ∇n2 v
|H (q)| ≥ , (30) |∇v| q where H is the mean curvature of S. Proof. Let / denote the Laplacian with respect to the metric induced on S. We have at q: 2 v = ∇n2 v + H ∇n v + / v ≤ ∇n v + H ∇n v,
since / v ≤ 0 there, and since, without loss of generality, we may take n = ∇v/ |∇v|. Thus, we obtain v ≤ ∇n2 v + H |∇v| , and (30) follows,
The right-hand side of (30) is easily recognized as the mean curvature of the level set of v at q. Thus this lemma is simply another version of the familiar geometric fact that when two surfaces are tangent at q and one lies entirely on one side from the other, then their mean curvature at q are correspondingly ordered. We prefer the statement in the lemma since it simplifies some of the explicit computations below. A first application is the following lemma. Lemma 4. Let p0 = (0, 0, 0), p1 = (0, 0, 1), p2 = (0, 0, −1) ∈ R3 , and let 0 < ε < 1/3. Then, for any compact surface S ⊂ B0 (3) \ D(ε) such that ∂S ⊂ ∂D(ε), there holds: sup |H | ≥ S
1 , 6
where H is the mean curvature of S. Proof. We consider two cases: (i) maxS v > 0; and (ii) v ≤ 0 on S. For case (i), take v = x 2 + y 2 − z2 /2. Let q ∈ S be such that v(q) = maxS v. Since v < 0 on ∂D(ε) we conclude that q ∈ / ∂S. Since v = 3, ∇n2 v ≤ 2, and |∇v| ≤ 2r < 6, Lemma 3 now yields: H (q) ≥
1 . 6
Now in case (ii) note that since S is smooth, it is contained in the double cone v < 0. Without loss of generality S1 = S∩{z > 0} = ∅, and we now take v = r12 = x 2 +y 2 +(z−1)2 , and let q as above be such that v(q) = maxS v. If q ∈ ∂S1 ⊂ ∂B1 (ε), then S1 ⊂ ∂B1 (ε) and H = 2/ε ≥ 6 at every interior point of S. On the other hand, if q ∈ / ∂S1 , then since w = 6, ∇n2 w = 2, and |∇w| = 2r1 , we obtain from Lemma 3 that H (q) ≥
2 1 > . r1 2
On a Penrose Inequality with Charge
721
Proposition 5. If m is sufficiently small, then for each ε > 0 there is η > 0 such that if φ − 1 C 2,α < η, then any closed surface S ⊂ M which is minimal in the conformal −1
perturbation (M, φ 4 g, φ −6 E) of (M, g, E) is contained in D(ε). Proof. The proof is established in three stages. We first show that S cannot enter the region outside B0 (3). We then do the same for the twice-perforated ball B0 (3) \ D(1/4). Finally, we prove the result in each of the two balls B1 (1/4) and B2 (1/4) separately. We will use the Euclidean metric δ, the Majumdar-Papapetrou metric g = u4 δ, and also its perturbation g˜ = φ 4 g. In order to avoid confusion we will use the dot product to denote the inner product with respect to δ, and indicate other metric objects by subscripts. We denote ν = φu, and note that Hg˜ = divg˜ (ng˜ ) =
1 1 4 divδ (ν 4 nδ ) = 2 Hδ + g(∇ν, ˜ ng˜ ), ν6 ν ν
(31)
where Hg˜ and Hδ denote the mean curvatures of S in the metrics g˜ and δ respectively. Suppose first that maxS r ≥ 3, where r is the Euclidean distance from p0 . Then, in view of δ r = 2/r, |∇r|δ = 1, ∇ 2 r = 0, we have according to Lemma 3 that at the point q with maximum r: |Hδ (q)| ≥
2 . r
Now, u2 = 1 + m/r1 + m/r2 , hence outside B0 (3), we have:
1 ∇u2 δ 1 m(1/r12 + 1/r22 ) m 3m |∇ log u|g = ≤ ≤ ≤ . 2 u4 2 (1 + m/r1 + m/r2 )2 (r − 1)2 4r Thus, using (31) and |φ − 1| , r 2 |∇φ|g ≤ φ − 1 C 2,α < η, we can estimate: −1
Hg˜ (q) ≥ |Hδ | − 4 |∇ log ν| g˜ ν2 4 2 − 2 |∇ log φ|g + |∇ log u|g ≥ r(1 + 2m/3)(1 + η)2 φ 2 η 4 3m ≥ − + r(1 + 2m/3)(1 + η)2 (1 − η)3 r 2 4r 2 1 4η + 3m ≥ . − r (1 + 2m/3)(1 + η)2 2(1 − η)3
Clearly if m and η are small enough, then Hg˜ (q) > 0, a contradiction. We conclude that S ⊂ B0 (3). Suppose now that S enters B0 (3) \ D(1/4). Then a similar estimate yields a point q in that region where 2m η |Hδ (q)| ≤ 4 1 + + 16m . 3 9(1 − η) Hence, if η and m are small enough, then we have |Hδ (q)| < 1/6 in contradiction to Lemma 4.
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G. Weinstein, S. Yamada
Therefore, we can now fix m and η0 small enough such that if η < η0 then S must lie in D(1/4). Consider the closed surface S1 = S ∩ B1 (1/4) with Hg˜ = 0. As above, we can estimate
Hg ≤ 4 |∇ log φ| < g
4η . 1−η
We will now apply Lemma 3 to the surface S1 and the function r1 in B1 (1/4) equipped with the metric g. Let q be the point where r1 (q) = maxS1 r1 < 1/4. We compute at q, using ng = ∇r1 / |∇r1 |g : 1 1 g r1 = 6 divδ (u2 ∇r1 ) = 6 u u
2u2 2 + ∇u · ∇r1 , r1
∇n2g r1 = ∇ng |∇r1 | = ∇ng u−2 = −
g(∇u2 , ∇r1 ) ∇u2 · ∇r1 . =− 4 u |∇r1 | u6
Thus, using m < 2, we can estimate: ∇r1 − ∇n2g r1 |∇r1 |
2u2 + 2∇u2 · ∇r1 r1
1 2u2 1 ∇r1 · ∇r2 = 4 − 2m 2 + u r1 r1 r22 1 2 ≥ 4 − 2m u r1 r1 ≥ . (1 + m)2
=
1 u4
We now obtain from Lemma 3:
4η r1 (q) ≤ Hg (q) ≤ . 2 (1 + m) 1−η Therefore, with m fixed, we see that maxS1 r1 → 0 as η → 0. The same argument can be applied to S2 = S ∩ B2 (1/4). This proves Proposition 5 Proposition 6. Let λ > 1. Then for m and η sufficiently small, the mass m of g, ˜ the area satisfy A˜ of the outermost horizon in g, ˜ and the charge Q m ≤ 2λm,
A˜ ≤ 8πλ2 m2 ,
≥ Q/λ. Q
(32)
is Proof. The metric of the perturbed space is g˜ = φ 4 u4 δ, therefore its mass m ∂φ 1 lim dA0 , m = 2m − 2π r→∞ Sr ∂r where A0 is the area element of Sr in the flat metric δ. Since ∂φ/∂r can be estimated on Sr by r −2 u2 φ C 2,α < r −2 u2 η, we find −1
η m ≤ 2m 1 + . m
On a Penrose Inequality with Charge
723
≥ Q/λ follows from (29). Thus, m ≤ 2λm, provided η ≤ m(λ − 1). Similarly Q We now note that g˜ admits one horizon. Indeed, the surface 0 = {s = 0} cutting the neck at its midpoint is totally geodesic, since it is the fixed-point set of the isometry sending any point p on one side of it to the corresponding point on the other. In particular, this surface is minimal and encloses the end ∞− , hence it is a horizon. Now let S be the outermost horizon. Then S is outer minimizing. According to Proposition 5, if m > 0 and η > 0 are sufficiently small, then S ⊂ D(ε), where ε ≤ 4η(1 + m)2 /(1 − η). Thus, ∂D(ε) encloses S, and we conclude: A˜ = Ag˜ (S) ≤ Ag˜ ∂D(ε) = φ 2 u4 dA0 ∂D(ε)
≤ (1 + η)
u dA0 ≤ 8πm (1 + η)
4
4
2
∂D(ε)
≤ 8π m (1 + η) 2
4
4η(1 + m)3 1+ m(1 − η)
2
4
1 1+ε 1+ m
2
.
Thus, with m > 0 fixed and small enough, we can choose η > 0 small enough to satisfy (32). Acknowledgement. We wish to thank Greg Galloway, Robert Hardt, and Robert Wald for useful discussions on this paper. We thank the American Institute of Mathematics for its hospitality. The first author also thanks the Erwin Schr¨odinger Institute for its hospitality.
References 1. Bartnik, R.: Existence of maximal surfaces in asymptotically flat spacetimes. Commun. Math. Phys. 94(2), 155–175 (1984) 2. Bray, H.: Proof of the Riemannian Penrose conjecture using the positive mass theorem. J. Differ. Geom. 59, 177–267 (2001) 3. Chaljub-Simon, A., Choquet-Bruhat, Y.: Probl`emes elliptiques du second ordre sur une vari´et´e euclidienne a` l’infini. Ann. Fac. Sci. Toulouse Math. (5) 1(1), 9–25 (1979) 4. Gibbons, G.W.: The Isoperimetric and Bogomolny Inequalities for Black Holes. In: Global Riemannian Geometry, T.J. Willmore, N. Hitchin (eds.), New York: John Wiley & Sons, New York, 1984 5. Gilbarg, D., Truginger, N.S.: Elliptic Partial Differential Equations of Second Order. Second Edition, New York: Springer-Verlag, 1983 6. Huisken, G., Ilmanen, T.: The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality. J. Differ. Geom. 59, 353–437 (2001) 7. Isenberg, J., Mazzeo, R., Pollack, D.: Gluing and Wormholes for the Einstein Constraint Equations. Commun. Math. Phys. 231, 529–568 (2002) ´ Murchadha, N., York, J.W. Jr.: Initial-Value Problem of General Relativity. III. Cou8. Isenberg, J., O pled Fields and the Scalar-Tensor Theory. Phys. Rev. D. 13(6), 1532–1537 (1976) 9. Jang, P.S.: Note on Cosmic Censorship. Phys. Rev. D. 20(4), 834–838 (1979) 10. Lee, J.M., Parker, T.H.: The Yamabe Problem. Bull. of the AMS 17(1), 37–92 (1987) 11. Meeks III, W., Simon, L., Yau, S.T.: Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann. Math. 16(3), 621–659 (1982) 12. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979) 13. Smith, B., Weinstein, G.: Quasiconvex foliations and asymptotically flat metrics of non-negative scalar curvature. Commun. Anal. Geom. 12(3), 511–551 (2004) 14. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3) 381–402 (1981) Communicated by G. W. Gibbons
Commun. Math. Phys. 257, 725–771 (2005) Digital Object Identifier (DOI) 10.1007/s00220-005-1330-9
Communications in
Mathematical Physics
Symmetry Classes of Disordered Fermions P. Heinzner1 , A. Huckleberry1 , M.R. Zirnbauer2 1 2
Fakult¨at f¨ur Mathematik, Ruhr-Universit¨at Bochum, Germany. E-mail: [email protected]; [email protected] Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Germany. E-mail: [email protected]
Received: 10 June 2004 / Accepted: 9 December 2004 Published online: 4 May 2005 – © Springer-Verlag 2005
Abstract: Building upon Dyson’s fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V , a space of endomorphisms in End(V ⊕ V ∗ ), a bilinear form on V ⊕ V ∗ which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V ∗ . Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago. 1. Introduction In a famous and influential paper published in 1962 (“The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics” [D]), Freeman J. Dyson classified matrix ensembles by a scheme that became fundamental to several areas of theoretical physics, including the statistical theory of complex many-body systems, mesoscopic physics, disordered electron systems, and the area of quantum chaos. Being set in the context of standard quantum mechanics, Dyson’s classification asserted that “the most general matrix ensemble, defined with a symmetry group that may be completely arbitrary, reduces to a direct product of independent irreducible ensembles
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each of which belongs to one of three known types.” These three ensembles, or rather their underlying matrix spaces, are nowadays known as the Wigner-Dyson symmetry classes of orthogonal, unitary, and symplectic symmetry. Over the last ten years, various matrix spaces beyond Dyson’s threefold way have come to the fore in random-matrix physics and mathematics. On the physics side, such spaces arise in problems of disordered or chaotic fermions; among these are the Euclidean Dirac operator in a stochastic gauge field background [V2], and quasiparticle excitations in disordered superconductors or metals in proximity to a superconductor [A2]. In the mathematical research area of number theory, the study of statistical correlations in the values of the Riemann zeta function, and more generally of families of L-functions, has prompted some of the same extensions [K]. A brief account of why new structures emerge on the physics side is as follows. When Dirac first wrote down his famous equation in 1928, he assumed that he was writing an equation for the wavefunction of the electron. Later, because of the instability caused by negative-energy solutions, the Dirac equation was reinterpreted (via second quantization) as an equation for the fermionic field operators of a quantum field theory. A similar change of viewpoint is carried out in reverse in the Hartree-Fock-Bogoliubov mean-field description of quasiparticle excitations in superconductors. There, one starts from the equations of motion for linear superpositions of the electron creation and annihilation operators, and reinterprets them as a unitary quantum dynamics for what might be called the quasiparticle ‘wavefunction’. In both cases – the Dirac equation and the quasiparticle dynamics of a superconductor – there enters a structure not present in the standard quantum mechanics underlying Dyson’s classification: the fermionic field operators are subject to a set of conditions known as the canonical anticommutation relations, and these are preserved by the quantum dynamics. Therefore, whenever second quantization is undone (assuming it can be undone) to return from field operators to wavefunctions, the wavefunction dynamics is required to preserve some extra structure. This puts a linear constraint on the allowed Hamiltonians. A good viewpoint to adopt is to attribute the extra invariant structure to the Hilbert space, thereby turning it into a Nambu space. It was conjectured some time ago [A2] that extending Dyson’s classification to the Nambu space setting, the relevant objects one is led to consider are large families of symmetric spaces of compact type. Past understanding of the systematic nature of the extended classification scheme relied on the mapping of disordered fermion problems to field theories with supersymmetric target spaces [Z] in combination with renormalization group ideas and the classification theory of Lie superalgebras. An extensive review of the mathematics and physics of symmetric spaces, covering the wide range from the basic definitions to various random-matrix applications, has recently been given in [C]. That work, however, offers no answers to the question as to why symmetric spaces are relevant for symmetry classification, and under what assumptions the classification by symmetric spaces is complete. In the present paper, we get to the bottom of the subject and, using a minimal set of tools from linear algebra, give a rigorous answer to the classification problem for disordered fermions. The rest of this introduction gives an overview of the mathematical model to be studied and a statement of our main result. We begin with a finite- or infinite-dimensional Hilbert space V carrying a unitary representation of some compact Lie group G0 – this is the group of unitary symmetries of the disordered fermion system. We emphasize that G0 need not be connected; in fact, it might be just a finite group.
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Let W = V ⊕ V ∗ , called the Nambu space of fermionic field operators, be equipped with the induced G0 -representation. This means that V is equipped with the given representation, and g(f ) := f ◦ g −1 for f ∈ V ∗ , g ∈ G0 . Let C : W → W be the C-antilinear involution determined by the Hermitian scalar product , V on V. In physics this operator is called particle-hole conjugation. Another canonical structure on W is the symmetric complex bilinear form b : W × W → C defined by b(v1 + f1 , v2 + f2 ) := f1 (v2 ) + f2 (v1 ) . It encodes the canonical anticommutation relations for fermions, and is related to the unitary structure , of W by b(w1 , w2 ) = Cw1 , w2 for all w1 , w2 ∈ W. It is assumed that G0 is contained in a group G – the total symmetry group of the fermion system – which is acting on W by transformations that are either unitary or antiunitary. An element g ∈ G either stabilizes V or exchanges V and V ∗ . In the latter case we say that g ∈ G mixes, and in the former case we say that it is nonmixing. The group G is generated by G0 and distinguished elements gT which act as antiunitary operators T : W → W. These are referred to as distinguished ‘time-reversal’ symmetries, or T -symmetries for short. The squares of the gT lie in the center of the abstract group G; we therefore require that the antiunitary operators T representing them satisfy T 2 = ±Id. The subgroup G0 is defined as the set of all elements of G which are represented as unitary, nonmixing operators on W. If T and T1 are distinguished time-reversal operators, then P := T T1 is a unitary symmetry. P may be mixing or nonmixing. In the latter case, P is in G0 . Therefore, modulo G0 , there exist at most two different T -symmetries. If there are exactly two such symmetries, we adopt the convention that T is mixing and T1 is nonmixing. Furthermore, it is assumed that T and T1 either commute or anticommute, i.e., T1 T = ±T T1 . As explained throughout this article, all of these situations are well motivated by physical considerations and examples. We note that time-reversal symmetry (and all other T -symmetries) of the disordered fermion system may also be broken; in this case T and T1 are eliminated from the mathematical model and G0 = G. Given W and the representation of G on it, the object of interest is the real vector space H of C-linear operators in End(W) that preserve the canonical structures b and , of W and commute with the G-action. Physically speaking, H is the space of ‘good’ Hamiltonians: the field operator dynamics generated by H ∈ H preserves both the canonical anticommutation relations and the probability in Nambu space, and is compatible with the prescribed symmetry group G. When unitary symmetries are present, the space H decomposes by blocks associated with isomorphism classes of G0 -subrepresentations occurring in W. To formalize this, recall that two unitary representations ρi : G0 → U(Vi ), i = 1, 2, are equivalent if and only if there exists a unitary C-linear isomorphism ϕ : V1 → V2 so that ˆ 0 denote the space ρ2 (g)(ϕ(v)) = ϕ(ρ1 (g)(v)) for all v ∈ V1 and for all g ∈ G0 . Let G ˆ0 of equivalence classes of irreducible unitary representations of G0 . An element λ ∈ G is called an isomorphism class for short. By standard facts (recall that every representation of a compact group is completely reducible) the unitary G0 -representation on V decomposes as an orthogonal sum over isomorphism classes: V = ⊕λ Vλ . The subspaces Vλ are called the G0 -isotypic components of V. Some of them may be zero. (Some of the isomorphism classes of G0 may just not be realized in V.)
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For simplicity suppose now that there is only one distinguished time-reversal symˆ 0 with Vλ = 0, consider the vector space T (Vλ ). If metry T , and for any fixed λ ∈ G T is nonmixing, i.e., T : V → V, then T (Vλ ) ⊂ V must coincide with the isotypic component for the same or some other isomorphism class. (Since conjugation by gT is an automorphism of G0 , the decomposition into G0 -isotypic components is preserved ˆ 0. by T .) If T is mixing, i.e., T : V → V ∗ , then T (Vλ ) = Vλ∗ , still with some λ ∈ G Now define the block Bλ to be the smallest G-invariant space containing Vλ ⊕ Vλ∗ . Note that if we are in the situation of nonmixing and T (Vλ ) = Vλ , then ∗ Bλ = Vλ ⊕ T (Vλ ) ⊕ Vλ ⊕ T (Vλ ) . On the other hand, if we are in the situation of mixing and T (Vλ ) = Vλ∗ , then Bλ = Vλ ⊕ T (Vλ∗ ) ⊕ Vλ∗ ⊕ T (Vλ ) . The block Bλ is halved if T (Vλ ) = Vλ resp. T (Vλ ) = Vλ∗ . Note that if there are two distinguished T -symmetries, the above discussion is only slightly more complicated. In any case we now have the basic G-invariant blocks Bλ . Because different blocks are built from representations of different isomorphism classes, the good Hamiltonians do not mix blocks. Thus every H ∈ H is a direct sum over blocks, and the structure analysis of H can be carried out for each block Bλ separately. If Vλ is infinite-dimensional, then to have good mathematical control we truncate to a finite-dimensional space Vλ ⊂ Vλ and form the associated block Bλ ⊂ W. The truncation is done in such a way that Bλ is a G-representation space and is Nambu. The goal now is to compute the space of Hermitian operators on Bλ which commute with the G-action and respect the canonical symmetric C-bilinear form b induced from that on V ⊕ V ∗ ; such a space of operators realizes what is called a symmetry class. For this, certain spaces of G0 -equivariant homomorphisms play an essential role, i.e., linear maps S : V1 → V2 between G0 -representation spaces which satisfy ρ2 (g) ◦ S = S ◦ ρ1 (g) for all g ∈ G0 , where ρi : G0 → U(Vi ), i = 1, 2, are the respective representations. If it is clear which representations are at hand, we often simply write g ◦ S = S ◦ g or S = gSg −1 . Thus we regard the space HomG0 (V1 , V2 ) of equivariant homomorphisms as the space of G0 -fixed vectors in the space Hom(V1 , V2 ) of all linear maps. If V1 = V2 = V , then these spaces are denoted by EndG0 (V ) and End(V ) respectively. Roughly speaking, there are two steps for computing the relevant spaces of Hermitian operators. First, the block Bλ is replaced by an analogous block Hλ of G0 -equivariant homomorphisms from a fixed representation space Rλ of isomorphism class λ and/or its dual Rλ∗ to Bλ . The space Hλ carries a canonical form (called either s or a) which is induced from b. As the notation indicates, although the original bilinear form on Bλ is symmetric, this induced form is either symmetric or alternating. Change of parity occurs in the most interesting case when there is a G0 -equivariant isomorphism ψ : Rλ → Rλ∗ . In that case there exists a bilinear form Fψ : Rλ × Rλ → C defined by Fψ (r, t) = ψ(r)(t), which is either symmetric or alternating. In a certain sense the form b is a product of Fψ and a canonical form on Hλ . Thus, if Fψ is alternating, then the canonical form on Hλ must also be alternating. After transferring to the space Hλ , in addition to the canonical bilinear form s or a we have a unitary structure and conjugation by one or two distinguished time-reversal symmetries. Such a symmetry T may be mixing or not, and both T 2 = Id and T 2 = −Id
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are possible. The second main step of our work is to understand these various cases, each of which is directly related to a classical symmetric space of compact type. Such are given by a classical Lie algebra g which is either sun , usp2n , or son (R). In the notation of symmetric spaces we have the following situation. Let g be the Lie algebra of antihermitian endomorphisms of Hλ which are isometries (in the sense of Lie algebra elements) of the induced complex bilinear form b = s or b = a. This is of compact type, because it is the intersection of the Lie algebra of the unitary group of Hλ and the complex Lie algebra of the group of isometries of b. Conjugation by the antiunitary mapping T defines an involution θ : g → g. The good Hamiltonians (restricted to the reduced block Hλ ) are the Hermitian operators h ∈ ig such that at the level of group action the one-parameter groups e−ith satisfy T e−ith = e+ith T , i.e., ih ∈ g must anticommute with T . Equivalently, if g = k ⊕ p is the decomposition of g into θ -eigenspaces, the space of operators which is to be computed is the (−1)-eigenspace p. The space of good Hamiltonians restricted to Hλ then is ip. Since the appropriate action of the Lie group K (with Lie algebra k) on this space is just conjugation, one identifies ip with the tangent space g/k of an associated symmetric space G/K of compact type. It should be underlined that there is more than one symmetric space associated to a Cartan decomposition g = k ⊕ p. We are most interested in the one consisting of the physical time-evolution operators e−ith ; if G (not to be confused with the symmetry group G) is the semisimple and simply connected Lie group with Lie algebra g, this is given as the image of the compact symmetric space G/K under the Cartan embedding into G defined by gK → gθ (g)−1 , where θ : G → G is the induced group involution. The following mathematical result is a conseqence of the detailed classification work in Sects. 3, 4 and 5. Theorem 1.1. The symmetric spaces which occur under these assumptions are irreducible classical symmetric spaces g/k of compact type. Conversely, every irreducible classical symmetric space of compact type occurs in this way. We emphasize that here the notion symmetric space is applied flexibly in the sense that depending on the circumstances it could mean either the infinitesimal model g/k or the Cartan-embedded compact symmetric space G/K. Theorem 1.1 settles the question of symmetry classes in disordered fermion systems; in fact every physics example is handled by one of the situations above. The paper is organized as follows. In Sect. 2, starting from physical considerations we motivate and develop the model that serves as the basis for subsequent mathematical work. Section 3 proves a number of results which are used to eliminate the group of unitary symmetries G0 . The main work of classification is given in Sects. 4 and 5. In Sect. 4 we handle the case where at most one distinguished time-reversal operator is present, and in Sect. 5 the case where there are two. There are numerous situations that must be considered, and in each case we precisely describe the symmetric space which occurs. Various examples taken from the physics literature are listed in Sect. 6, illustrating the general classification theory. 2. Disordered Fermions with Symmetries ‘Fermions’ is the physics name for the elementary particles which all matter is made of. The goal of the present article is to establish a symmetry classification of Hamiltonians which are quadratic in the fermion creation and annihilation operators. To motivate
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this restriction, note that any Hamiltonian for fermions at the fundamental level is of Dirac type; thus it is always quadratic in the fermion operators, albeit with time-dependent coefficients that are themselves operators. At the nonrelativistic or effective level, quadratic Hamiltonians arise in the Hartree-Fock mean-field approximation for metals and the Hartree-Fock-Bogoliubov approximation for superconductors. By the LandauFermi liquid principle, such mean-field or noninteracting Hamiltonians give an adequate description of physical reality at very low temperatures. In the present section, starting from a physical framework, we develop the appropriate model that will serve as the basis for the mathematical work done later on. Please be advised that disorder, though advertised in the title of the section and in the title of paper, will play no explicit role here. Nevertheless, disorder (and/or chaos) are the indispensable agents that must be present in order to remove specific and nongeneric features from the physical system and make a classification by basic symmetries meaningful. In other words, what we carry out in this paper is the first step of a two-step program. This first step is to identify in the total space of Hamiltonians some linear subspaces that are relevant (in Dyson’s sense) from a symmetry perspective. The second step is to put probability measures on these spaces and work out the disorder averages and statistical correlation functions of interest. It is this latter step that ultimately justifies the first one and thus determines the name of the game. 2.1. The Nambu space model for fermions. The starting point for our considerations is the formalism of second quantization. Its relevant aspects will now be reviewed so as to introduce the key physical notions as well as the proper mathematical language. Let i = 1, 2, . . . label an orthonormal set of quantum states for a single fermion. Second quantizing the many-fermion system means to associate with each i a pair of operators ci† and ci , which are called fermion creation and annihilation operators, respectively, and are related to each other by an operation of Hermitian conjugation † : ci → ci† . They are subject to the canonical anticommutation relations ci† cj† + cj† ci† = 0 ,
ci cj + cj ci = 0 ,
ci† cj + cj ci† = δij ,
(2.1)
for all i, j . They act in a Fock space, i.e., in a vector space with a distinguished vector, called the ‘vacuum’, which is annihilated by all of the operators ci (i = 1, 2, . . . ). Applying n creation operators to the vacuum one gets a state vector for n fermions. A field operator ψ is a linear combination of creation and annihilation operators, † ψ= vi c i + f i ci , i
with complex coefficients vi and fi . To put this in mathematical terms, let V be the complex Hilbert space of single-fermion states. (We do not worry here about complications due to the dimension of V being infinite. Later rigorous work will be carried out in the finite-dimensional setting.) Fock space then is the exterior algebra ∧V = C ⊕ V ⊕ ∧2 V ⊕ . . . , with the vacuum being the one-dimensional subspace of constants. Creating a single fermion amounts to exterior multiplication by a vector v ∈ V and is denoted by ε(v) : ∧n V → ∧n+1 V. To annihilate a fermion, one contracts with an element f of the dual
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space V ∗ . In other words, one applies the antiderivation ι(f ) : ∧n V → ∧n−1 V given by ι(f ) · 1 = 0, ι(f ) v = f (v), ι(f ) (v1 ∧ v2 ) = f (v1 ) v2 − f (v2 ) v1 , etc. In that mathematical framework the canonical anticommutation relations read ε(v)ε(v) ˜ + ε(v)ε(v) ˜ =0, ˜ ˜ ι(f )ι(f ) + ι(f )ι(f ) = 0 , ι(f )ε(v) + ε(v)ι(f ) = f (v) .
(2.2)
They can be viewed as the defining relations of an associative algebra, the so-called Clifford algebra C(W), which is generated by the vector space W := V ⊕ V ∗ over C. This vector space W is sometimes referred to as Nambu space in physics. Since we only consider Hamiltonians that are quadratic in the creation and annihilation operators, we will be able to reduce the second-quantized formulation on ∧V to standard single-particle quantum mechanics, albeit on the Nambu space W carrying some extra structure. Note that W is isomorphic to the space of field operators ψ. On W = V ⊕ V ∗ there exists a canonical symmetric C-bilinear form b defined by (fi v˜i + f˜i vi ) . b(v + f, v˜ + f˜) = f (v) ˜ + f˜(v) = i
The significance of this bilinear form in the present context lies in the fact that it encodes on W the canonical anticommutation relations (2.1), or (2.2). Indeed, we can view a † ∗ field operator ψ = i (vi ci + fi ci ) either as a vector ψ = v + f ∈ V ⊕ V , or equivalently as a degree-one operator ψ = ε(v) + ι(f ) in the Clifford algebra acting on ∧V. Adopting the operator perspective, we get from (2.2) that ˜ = f (v) fi v˜i + f˜i vi . ψ ψ˜ + ψψ ˜ + f˜(v) = i
˜ Thus Switching to the vector perspective we have the same answer from b(ψ, ψ). ˜ = b(ψ, ψ) ˜ . ψ ψ˜ + ψψ Definition 2.1. In the Nambu space model for fermions one identifies the space of field operators ψ with the complex vector space W = V ⊕ V ∗ equipped with its canonical unitary structure , and canonical symmetric complex bilinear form b. Remark. Having already expounded the physical origin of the symmetric bilinear form b, let us now specify the canonical unitary structure of W. The complex vector space V, being isomorphic to the Hilbert space of single-particle states, comes with a Hermitian scalar product (or unitary structure) , V . Given , V define a C-antilinear bijection C : V → V ∗ by Cv = v, ·V , and extend this to an antilinear transformation C : W → W by the requirement C 2 = Id. Thus C|V ∗ = (C|V )−1 . The operator C is called particle-hole conjugation in physics. Using C, transfer the unitary structure from V to V ∗ in the natural way: f, f˜V ∗ := Cf, C f˜V = C f˜, Cf V . The canonical unitary structure of W is then given by v¯i v˜i + f¯i f˜i . v + f, v˜ + f˜ = v, v ˜ V + f, f˜V ∗ = i
Thus , is the orthogonal sum of the Hermitian scalar products on V and V ∗ .
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Proposition 2.2. The canonical unitary structure and symmetric complex bilinear form of W are related by ˜ = b(C ψ, ψ) ˜ . ψ, ψ Proof. Given an orthonormal basis c1† , c1 , c2† , c2 , . . . this is immediate from (vi ci† + fi ci ) = (v¯i ci + f¯i ci† ) C i
i
and the expressions for , and b in components.
Returning to the physics way of telling the story, consider the most general Hamiltonian H which is quadratic in the single-fermion creation and annihilation operators. Assuming H to be Hermitian, using the canonical anticommutation relations (2.1), and omitting an additive constant (which is of no consequence in physics) this has the form H = 21 Aij ci† cj − cj ci† + 21 Bij ci† cj† + B¯ ij cj ci , ij
ij
where Aij = A¯ j i (from H = H † ) and Bij = −Bj i (from ci cj = −cj ci ). The Hamiltonians H act on the field operators ψ by the commutator, ψ → [H, ψ] ≡ H ψ − ψH , and the time evolution is determined by the Heisenberg equation of motion, −i
dψ = [H, ψ] , dt
with being Planck’s constant. By the canonical anticommutation relations, this dynamical equation is equivalent to a system of linear differential equations for the coefficients vi and fi : Aij vj + Bij fj , −iv˙i = j ˙ ifi = B¯ ij vj + A¯ ij fj . j
If these are assembled into a column vector v, the evolution equation takes the form i A B v˙ = Xv , X = . −B¯ −A¯ To recast all this in concise terms, we need some further mathematical background. Notwithstanding the fact that in practice we always consider the Fock space representation C(W) → End(∧V) by w = v + f → ε(v) + ι(f ), it should be stated that the primary (or universal) definition of the Clifford algebra C(W) is as the associative algebra generated by W ⊕ C with relations w1 w2 + w2 w1 = b(w1 , w2 )
(w1 , w2 ∈ W) .
(2.3)
The Clifford algebra is graded by C(W) = C 0 (W) ⊕ C 1 (W) ⊕ C 2 (W) ⊕ . . . , where C 0 (W) ≡ C, C 1 (W) ∼ = W, and C n (W) for n ≥ 2 is the linear space of skew-symmetrized degree-n monomials in the elements of W. In particular, C 2 (W) is the linear space of skew-symmetric quadratic monomials w1 w2 − w2 w1 (w1 , w2 ∈ W).
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From the Clifford algebra perspective, a quadratic Hamiltonian H is viewed as an operator in the degree-two component C 2 (W). Let us therefore gather some standard facts about C 2 (W). First among these is that C 2 (W) is a complex Lie algebra with the commutator playing the role of the Lie bracket (an exposition of this fact for the case of a Clifford algebra over R is found in [B3]; the complex case is no different). Second, in addition to acting on itself by the commutator, the Lie algebra C 2 (W) acts (still by the commutator) on all of the components C k (W) of degree k ≥ 1 of the Clifford algebra C(W). In particular, C 2 (W) acts on C 1 (W). Third, C 2 (W) turns out to be canonically isomorphic to the complex orthogonal Lie algebra so(W, b) which is associated with the vector space W = V ⊕ V ∗ and its canonical symmetric complex bilinear form b; this Lie algebra so(W, b) is defined to be the subspace of elements E ∈ End(W) satisfying the condition b(Ew1 , w2 ) + b(w1 , Ew2 ) = 0
(for all w1 , w2 ∈ W) .
The canonical isomorphism C 2 (W) → so(W, b) is given by the commutator action of C 2 (W) on C 1 (W) ∼ = W, i.e., by sending a ∈ C 2 (W) to [a, ·] = E ∈ End(W); the latter indeed lies in so(W, b) as follows from the expression for b(Ew1 , w2 ) + b(w1 , Ew2 ) given by the canonical anticommutation relations (2.3), from the Jacobi identity [a, w1 ] w2 + w2 [a, w1 ] + w1 [a, w2 ] + [a, w2 ] w1 = [a, w1 w2 + w2 w1 ] , and from the fact that w1 w2 + w2 w1 lies in the center of the Clifford algebra. To describe so(W, b) explicitly, decompose the endomorphisms E ∈ End(V ⊕ V ∗ ) into blocks as AB E= , CD where A ∈ End(V), B ∈ Hom(V ∗ , V), C ∈ Hom(V, V ∗ ) and D ∈ End(V ∗ ). Let the adjoint (or transpose) of A ∈ End(V) be denoted by At ∈ End(V ∗ ), and call an element C in Hom(V, V ∗ ) skew if Ct = −C, i.e., if (Cv1 )(v2 ) = −(C v2 )(v1 ). AB Proposition 2.3. An endomorphism E = ∈ End(V ⊕ V ∗ ) lies in the complex CD orthogonal Lie algebra so(V ⊕ V ∗ , b) if and only if B, C are skew and D = −At . Proof. Consider first the case B = C = 0, and let D = −At . Then b E(v + f ), v˜ + f˜ = b(Av − At f, v˜ + f˜) = f˜(Av) − At f (v) ˜ t ˜ t ˜ = A f (v) − f (Av) ˜ = −b(v + f, Av˜ − A f ) = −b v + f, E(v˜ + f˜) . Using Bt = −B and Ct = −C, a similar calculation for the case A = 0 gives b E(v + f ), v˜ + f˜ = b(Bf + Cv, v˜ + f˜) = Cv(v) ˜ + f˜(Bf ) = −f (Bf˜) − Cv(v) ˜ = −b(v + f, Bf˜ + Cv) ˜ = −b v + f, E(v˜ + f˜) . Since these two cases complement each other, we see that the stated conditions on E ∈ End(W) are sufficient in order for E to be in so(W, b). The calculation can equally well be read backwards; thus the conditions are both sufficient and necessary.
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Let us now make the connection to physics, where C(W) is represented on Fock space and the elements v + f = w ∈ W become field operators ψ = ε(v) + ι(f ). Fixing orthonormal bases c1† , c2† , . . . of V and c1 , c2 , . . . of V ∗ as before, we assign matrices with matrix elements Aij , Bij , Cij to the linear operators A, B, C. A straightforward computation using the canonical anticommutation relations then yields: Proposition 2.4. The inverse of the Lie algebra automorphism C 2 (W) → so(W, b) is the C-linear mapping given by A B → 21 Aij (ci† cj − cj ci† ) + 21 (Bij ci† cj† + Cij ci cj ) . t C −A ij ij Now recall that C 2 (W) acts on the degree-one component C 1 (W) by the commutator. By the isomorphisms C 2 (W) ∼ = so(W, b) and C 1 (W) ∼ = W, this action coincides with the fundamental representation of so(W, b) on its defining vector space W. In other words, taking the commutator of the Hamiltonian H ∈ C 2 (W) with a field operator ψ ∈ C 1 (W) yields thesame answer as viewing H as an element of so(W, b), then A B to the vector ψ = v + f ∈ W by applying H = C −At H · (v + f ) = (Av + Bf ) + (Cv − At f ) , and finally reinterpreting the result as a field operator in C 1 (W). The closure relation [C 2 (W), C 1 (W)] ⊂ C 1 (W) and the isomorphism C 1 (W) ∼ =W make it possible to reduce the dynamics of field operators to a dynamics on the Nambu space W. After reduction, as we have seen, the generators X ∈ End(V ⊕ V ∗ ) of time evolutions of the physical system are of the special form i A B X= , ∗ t B −A where B ∈ Hom(V ∗ , V) is skew, and A = A∗ ∈ End(V) is self-adjoint w.r.t. , V . Proposition 2.5. The one-parameter groups of time evolutions t → etX in the Nambu space model preserve both the canonical unitary structure , and the canonical symmetric complex bilinear form b of W = V ⊕ V ∗ . Proof. By Prop. 2.3 the generator X is an element of the complex Lie algebra so(W, b). Hence the exponential Ut = etX lies in the complex orthogonal Lie group SO(W, b), which is defined to be the set of solutions g in End(W) of the conditions ˜ = b(ψ, ψ) ˜ , b(gψ, g ψ)
Det(g) = 1 .
and
Since A = A∗ , and B∗ ∈ Hom(V, V ∗ ) is the adjoint of B ∈ Hom(V ∗ , V), the generator X is antihermitian with respect to the unitary structure of W. The exponentiated generator Ut therefore lies in the unitary group U(W), which is to say that ˜ = ψ, ψ ˜ Ut ψ, Ut ψ for all real t. Thus Ut preserves both b and , .
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Remark. In physical language, the invariance of b under time evolutions means that the canonical anticommutation relations for fermionic field operators do not change with time. Invariance of , means that probability in Nambu space is conserved. (If the quadratic Hamiltonian H arises as the mean-field approximation to some many-fermion problem, the latter conservation law holds as long as quasiparticles do not interact and thereby are protected from decay into multi-particle states.) We now distill the essence of the information conveyed in this section. The quantum theory of many-fermion systems is set up in a Hilbert space called the fermionic Fock space in physics (or the spinor representation in mathematics). The field operators of the physical system span a vector space W = V ⊕ V ∗ , which generates a Clifford algebra C(W) whose defining relations are the canonical anticommutation relations. Since [C 2 (W), C 1 (W)] ⊂ C 1 (W), the discussion of the field operator dynamics for the important case of quadratic Hamiltonians H ∈ C 2 (W) can be reduced to a discussion on the Nambu space W ∼ = C 1 (W). Via this reduction, the vector space W inherits two natural structures: the canonical symmetric complex bilinear form b encoding the anticommutation relations, and a canonical unitary structure , determined by the Hermitian scalar product of V. Both of these structures are invariant, i.e., are preserved by physical time evolutions. Under the reduction to W, the commutator action of C 2 (W) on C 1 (W) becomes the fundamental representation of so(W, b) on W. 2.2. Symmetry groups. Following Dyson, the classification of disordered fermion systems will be carried out in a setting that prescribes two pieces of data: • One is given a Nambu space W = V ⊕ V ∗ equipped with its canonical unitary structure , and canonical symmetric C-bilinear form b. • On W there acts a group G of unitary and antiunitary operators (the joint symmetry group of a multi-parameter family of fermionic quantum systems). Given this setup, one is interested in the linear space of Hamiltonians H with the property that they commute with the G-action on W, while preserving the invariant structures b and , of W under time evolution by e−itH / . Such a space of Hamiltonians is of course reducible in general, i.e., the Hamiltonian matrices decompose into blocks. The goal of classification is to enumerate all the symmetry classes, i.e., all the types of irreducible blocks which occur in this way. In the present subsection we provide some information on what is meant by unitary and antiunitary symmetries in the present context. We begin by recalling the basic notion of a symmetry group in quantum Hamiltonian systems. In classical mechanics the symmetry group G0 of a Hamiltonian system is understood to be the group of symplectomorphisms that commute with the phase flow of the system. Examples are the rotation group for systems in a central field, and the group of Euclidean motions for systems with Euclidean invariance. In passing from classical to quantum mechanics, one replaces the classical phase space by a complex Hilbert space V, and assigns to the symmetry group G0 a (projective) representation by unitary C-linear operators on V. While the consequences due to one-parameter continuous subgroups of G0 are particularly clear from Noether’s theorem [A], the components of G0 not connected with the identity also play an important role. A prominent example is provided by the operator for space reflection. Its eigenspaces are the subspaces of states with positive and negative parity, and they reduce the matrix of any reflection-invariant Hamiltonian to two blocks.
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Not all symmetries of a quantum mechanical system are of the canonical, unitary kind: the prime counterexample is the operation gT of inverting the time direction – called time reversal for short. In classical mechanics this operation reverses the sign of the symplectic structure of phase space; in quantum mechanics its algebraic properties reflect the fact that the time t enters in the Dirac, Pauli, or Schr¨odinger equation as id/dt: there, time reversal gT is represented by an antiunitary operator T , which is to say that T is complex antilinear: T (zv) = z¯ T v
(z ∈ C, v ∈ V) ,
and preserves the Hermitian scalar product up to complex conjugation: ˜ V. v, v ˜ V = T v, T v Another example of such an operation is charge conjugation in relativistic theories. Further examples are provided by chiral symmetry transformations (see Sect. 2.3). By the symmetry group G of a quantum mechanical system with Hamiltonian H , one then means the group of all unitary and antiunitary transformations g of V that leave the Hamiltonian invariant: gH g −1 = H . It should be noted that finding the total symmetry group of a quantization of some Hamiltonian system is not always straightforward. The reason is that there may exist nonobvious quantum symmetries such as Hecke symmetries, which are of number-theoretic origin and have no classical limit. For our purposes, however, this complication will not be an issue. We take the group G and its action on the Hilbert space to be fundamental and given, and then ask what is the linear space of Hamiltonians that commute with the G-action. For technical reasons, we assume the group G0 to be compact; this is an assumption that covers most (if not all) of the cases of interest in physics. The noncompact group of space translations can be incorporated, if necessary, by wrapping the system around a torus, whereby translations are turned into compact torus rotations. What we have sketched – a symmetry group G acting on a Hilbert space V – is the framework underlying Dyson’s classification. As was explained in Sect. 2.1, we wish to enlarge it so as to capture all examples that arise in disordered fermion physics. For this, recall that in the Nambu space model for fermions, the Hilbert space is not V but the space of field operators W = V ⊕ V ∗ . The given G-representation on V therefore needs to be extended to a representation on W. This is done by the condition that the pairing between V and V ∗ (and thus the pairing between fermion creation and annihilation operators) be preserved. In other words, if U : V → V and A : V → V are unitary resp. antiunitary operators, their induced representations on V ∗ (which we still denote by the same symbols) are defined by requiring that (Uf )(U v) = f (v) = (Af )(Av) for all v ∈ V and f ∈ V ∗ . In particular the G0 -representation on V ∗ is the dual one, U (f ) = f ◦ U −1 . Equivalently, the G-representation on W is defined so as to be compatible with particle-hole conjugation C : W → W in the sense that operations commute: CU = U C ,
and CA = AC .
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Indeed, if f = Cv then f (v) ˜ = v, v ˜ and from the invariance of the pairing between V and V ∗ one infers the relations v, v ˜ = (Uf )(U v) ˜ = U −1 C −1 U Cv, v ˜ and v, v ˜ = −1 −1 (Af )(Av) ˜ = A C ACv, v. ˜ While the framework so obtained is flexible enough to capture the situations that arise in the nonrelativistic quasiparticle physics of disordered metals, semiconductors and superconductors, it is still slightly too narrow to accommodate some much studied examples that have emerged from elementary particle physics. Let us explain this. 2.3. The Euclidean Dirac operator. An important development in random-matrix physics over the last ten years was the formulation [V2] and study of the so-called chiral ensembles, which model Dirac fermions in a random gauge field background, and lie beyond Dyson’s 3-way classification. From the viewpoint of applications, these randommatrix models have the merit of capturing some universal features of the Dirac spectrum of quantum chromodynamics (QCD) in the low-energy limit. In the present subsection we will demonstrate that, but for one minor difference, they fit naturally into our fermionic Nambu space model with symmetries. Let M be a four-dimensional Euclidean space-time (more generally, M could be a Riemannian 4-manifold with spin structure), and consider over M a unitary spinor bundle S twisted by a module R for the action of some compact gauge group K. Denote by V the Hilbert space of L2 -sections of the twisted bundle S ⊗ R. Now let DA be a self-adjoint Dirac operator for V in a given gauge field background (or gauge connection) A. Although DA is not a Hamiltonian in the strict sense of the word, it has all the right mathematical attributes in the sense of Sect. 2.1; in particular it determines a Hermitian form, called the action functional, on differentiable sections ψ ∈ V. In physics notation this functional is written ¯ ψ → ψ(x) · (DA ψ) (x) d 4 x , DA = iγ µ (∂µ − Aµ ) , M
= γ (eµ ) are the gamma matrices [i.e., the Clifford action γ : T ∗ M → End(S) evaluated on the dual eµ of an orthonormal coordinate frame eµ of T M], the operators ∂µ are the partial derivatives corresponding to the eµ , and Aµ (x) ∈ Lie(K) are the where γ µ
components of the gauge field. If the physical situation calls for a mass, then one adds a complex number im (times the unit operator on V) to the expression for DA . The Dirac operators of prime interest to low-energy QCD have zero (or small) mass. To express the massless nature of DA one introduces an object called the chirality operator in mathematics [B3], or γ5 = γ 0 γ 1 γ 2 γ 3 in physics. = γ5 is a section of End(S) which is self-adjoint and involutory ( 2 = Id) and anticommutes with the Clifford action (γ µ + γ µ = 0). By the last property one has DA + DA = 0 in the massless limit. This relation is called chiral symmetry in physics. Note, however, that chiral ‘symmetry’ is not a symmetry in the sense of the present paper. (Symmetries always commute with the Hamiltonian, never do they anticommute with it!) Nonetheless, we shall now recognize chiral symmetry as being equivalent to a true symmetry, by importing the Dirac operator into the Nambu space model as follows. As before, take Nambu space to be the sum W = V ⊕ V ∗ equipped with its canonical unitary structure , and symmetric complex bilinear form b. The antilinear bijection C : V → V ∗ and C : V ∗ → V is still defined by w1 , w2 = b(Cw1 , w2 ).
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Now extend the Dirac operator DA ∈ iu(V) to an operator DA that acts diagonally on W = V ⊕ V ∗ , by requiring DA to satisfy the commutation law C iDA = iDA C, or equivalently CDA = −DA C. Thus, DA ∈ End(V) ⊕ End(V ∗ ) → End(W) , t . The diagonally extended operator D lies in the and DA on End(V ∗ ) is given by −DA A intersection of so(W, b) with iu(W) – as is required in order for the statement of Prop. 2.5 to carry over to the one-parameter group t → eit DA . The property that DA does not mix V and V ∗ can be attributed to the existence of a U1 symmetry group that has V and V ∗ as inequivalent representation spaces. To implement the chiral symmetry of the massless limit, extend the chirality operator to a diagonally acting endomorphism in End(V) ⊕ End(V ∗ ) by CC −1 = . The extended operators still satisfy the chiral symmetry relation DA + DA = 0. Then define an antiunitary operator T by T := C. Note that this is not the operation of reversing the time but will still be called the ‘time reversal’ for short. Because DA anticommutes with both C and , one has
T DA T −1 = DA . Thus T is a true symmetry of the (extended) Dirac operator in the massless limit. Note that CT = T C from C = C. As was announced above, the situation is the same as before but for one difference: while the time reversal in Sect. 2.2 was an operator T : V → V and T : V ∗ → V ∗ , the present one is an operator T : V → V ∗ and T : V ∗ → V. We refer to the latter type as mixing, and the former as nonmixing. To summarize, physical systems modelled by the Euclidean (or positive signature) Dirac operator are naturally incorporated into the framework of Sects. 2.1 and 2.2. The Hilbert space V here is the space of L2 -sections of a twisted spinor bundle over Euclidean space-time, and the role of the Hamiltonian is taken by the quadratic action functional of the Dirac fermion theory. When transcribed into the Nambu space W = V ⊕ V ∗ , the chiral ‘symmetry’ of the massless theory can be expressed as a true antiunitary symmetry T , with the only new feature being that T mixes V and V ∗ . The most general situation occurring in physics may exhibit, beside T , one or several other antiunitary symmetries. In the example at hand this happens if the representation space R carries a complex bilinear form which is invariant under gauge transformations (see Sects. 6.2.2 and 6.2.3 for the details). The Dirac operator DA then has one extra antiunitary symmetry, say T1 , which is nonmixing. Forming the composition of T1 with T we get a mixing unitary symmetry P = T T1 : V ↔ V ∗ . This fact leads us to adopt the final framework described in the next subsection. 2.4. The mathematical model. The following model is now well motivated. We are given a Nambu space (W, b, , ) carrying the action of a compact group G. The group G0 is defined to be the subgroup of G which acts by canonical unitary transformations, i.e., unitary transformations that preserve the decomposition W = V ⊕ V ∗ . The full symmetry group G is generated by G0 and at most two distinguished antiunitary time-reversal operators. If there is just one, we denote it by T , and if there are two, by T and T1 . In the latter case we adopt the convention that T mixes, i.e., T : V → V ∗ , while T1 is nonmixing. The distinguished time-reversal symmetries always satisfy T 2 = ±Id and T12 = ±Id. In the case that there are two, it is assumed that they commute or anticommute, i.e., T1 T = ±T T1 . Consequently the unitary operator P = T T1 (which
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mixes) also satisfies P 2 = ±Id. When P is present we let G1 denote the Z2 -extension of G0 defined by P and refer to it as the full group of unitary symmetries. We emphasize that the original action of G0 on V has been extended to W via its canonically induced action on V ∗ . In other words, if f ∈ V ∗ then g(f )(v) = f (g −1 (v)). This is equivalent to requiring that a unitary operator U ∈ G0 commutes with particlehole conjugation C : W → W. In fact we require that all operators of G commute with C. Whereas the unitary operators preserve the Hermitian scalar product , , for an antiunitary operator A we have that Aw1 , Aw2 = w1 , w2 for all w1 , w2 ∈ W. If U is an operator coming from G0 and T is a distinguished time-reversal symmetry, then T U T −1 is unitary and nonmixing, i.e., it is in G0 . Thus, for the corresponding operator gT in G, we assume that gT normalizes G0 and gT2 is in the center of G0 . According to Prop. 2.5 the time evolutions of the physical system leave the structure of Nambu space invariant. The infinitesimal version of this statement is that the Hamiltonians H lie in the intersection of the complex orthogonal Lie algebra so(W, b) with iu(W), the Hermitian operators on W. Let us summarize our situation in the language and notation introduced above. Definition 2.6. The data in the Nambu space model for fermions with symmetries is (W, b, , ; G), where the compact group G is called the symmetry group of the system. G is represented on W = V ⊕ V ∗ by unitary and antiunitary operators that preserve the structure of W; i.e., for every unitary U and antiunitary A one has ˜ = U ψ, U ψ ˜ = Aψ, Aψ ˜ , ψ, ψ
˜ = b(U ψ, U ψ) ˜ = b(Aψ, Aψ) ˜ b(ψ, ψ)
for all ψ, ψ˜ ∈ W. The space of ‘good’Hamiltonians is the R-vector space H of operators H in so(W, b) ∩ iu(W) that commute with the G-action: U H U −1 = H = AH A−1 . At the group level of time evolutions this means that U e−itH / = e−itH / U ,
Ae−itH / = e+itH / A ,
for all unitary U , antiunitary A, H ∈ H, and t ∈ R. We remind the reader that the subgroup of unitary operators which preserves the decomposition W = V ⊕ V ∗ is denoted by G0 , and the full group of unitaries by G1 . Several further remarks are in order. First, for a unitary U ∈ G1 (resp. antiunitary A), the compatibility of b with the G-action is a consequence of Prop. 2.2 and the commutation law CU = U C and CA = AC. Second, it is possible that the fermion system does not have any antiunitary symmetries and G = G0 . When some antiunitary symmetries are present, G is generated by G0 and one or at most two distinguished time-reversal symmetries as explained above. Third, motivated by the prime physics example of time reversal, we have assumed that the (one or two) distinguished time-reversal symmetries T satisfy T 2 = ±Id. The reason for this can be explained as follows. The operator T has been chosen to represent some kind of inversion symmetry. Since this means that conjugation by T 2 represents the unit operator, T 2 must be a unitary multiple of the identity on any subspace of W which is irreducible under time evolutions of the fermion system. Thus for all practical purposes we may assume that T is a projective involution, i.e., T 2 = z × Id with z a complex number of unit modulus.
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Proposition 2.7. If a projective involution T : W → W of a unitary vector space W is antiunitary, then either T 2 = +IdW or T 2 = −IdW . Proof. A projective involution T has square T 2 = z × Id with z ∈ C \ {0}. Since T is antiunitary, T 2 is unitary, and hence |z| = 1. But an antiunitary operator is C-antilinear, and therefore the associative law T 2 · T = T · T 2 forces z to be real, leaving only the possibilities T 2 = ±Id. Since this work is meant to simultaneously handle symmetry at both the Lie algebra and Lie group level, a final word should be said about the notion that a bilinear form F is respected by a transformation B.At the group level when B is invertible and is regarded as being in GL(W ), where W is the underlying vector space of F : W ×W → C, this means that B is an isometry in the sense that F (Bw1 , Bw2 ) = F (w1 , w2 ) for all w1 , w2 ∈ W . On the other hand, at the Lie algebra level where B ∈ End(W ), this means that for all d w1 , w2 ∈ W one has dt F (etB w1 , etB w2 )|t=0 = F (Bw1 , w2 ) + F (w1 , Bw2 ) = 0. 3. Reduction to the Case of G0 = {Id} Recall that our main goal, e.g., on the Lie algebra level, is to describe the space of G0 -invariant endomorphisms which on a block in Nambu space are compatible with the unitary structure, time reversal and the symmetric C-bilinear form. Here we prove results which allow us to transfer this space to a certain space of G0 -equivariant homomorphisms. The unitary structure, time reversal and the bilinear form are transferred canonically, and as before, compatibility with these structures is required. However, in the new setting G0 acts trivially. This is of course an essential simplification, and paves the way toward our classification goal. ˆ 0 denotes a fixed isomorphism class (i.e., an equivalence class In this section λ ∈ G of irreducible representations of G0 ), and λ∗ denotes its dual. A block is determined by a choice of finite-dimensional G0 -invariant subspace V = Vλ (in the given Hilbert space V) such that all of its irreducible subrepresentations have isomorphism class λ. The full group G of (unitary and antiunitary) symmetries is generated by G0 and at most two distinguished time-reversal symmetries. Throughout this section (and also in Sects. 4.2, 4.3, 5.1) we assume that these time-reversal operators T stabilize the truncated subspace W = V ⊕ V ∗ of Nambu space: TW = W . The case where one or both time-reversal symmetries do not stabilize W , i.e., where a larger block is generated, is handled in Sects. 4.4 and 5.2. 3.1. Spaces of equivariant homomorphisms. If , V is the initial unitary structure on V , one defines C : V → V ∗ by C(v)(w) = v, wV . Taking C|V ∗ to be the inverse of this map, one obtains the associated C-antilinear isomorphism C : W → W . All symmetries in G are assumed to commute with C. We remind the reader that G0 acts on V ∗ by g(f ) = f ◦ g −1 . Let R be a fixed irreducible G0 -representation space which is in λ. Denote by d its dimension. Of course R ∗ is a representative of λ∗ . We fix an antilinear bijection ι : R → R∗ ,
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which is defined by a G0 -invariant unitary structure , R on R. (Note the change of meaning of the symbol ι as compared to Sect. 2.1.) In the sequel we will often make use of the following consequence of Schur’s Lemma. (Note the change of meaning of the symbol ψ as compared to Sect. 2.1.) Proposition 3.1. If two irreducible G0 -representation spaces R1 and R2 are equivariantly isomorphic by ψ : R1 → R2 , then HomG0 (R1 , R2 ) = C · ψ, i.e., the linear space of G0 -equivariant homomorphisms from R1 to R2 has complex dimension one and every operator in it is some multiple of ψ. The following related statement was essential to Dyson’s classification and will play a similarly important role in the present article. Lemma 3.2. If an irreducible G0 -representation space R is equivariantly isomorphic to its dual R ∗ by an isomorphism ψ : R → R ∗ , then ψ is either symmetric or alternating, i.e., either ψ(r)(t) = ψ(t)(r) or ψ(r)(t) = −ψ(t)(r) for all r, t ∈ R. Proof. It is convenient to think of ψ as defining an invariant bilinear form B(r, t) = ψ(r)(t) on R. We then decompose B into its symmetric and alternating parts, B = S +A, where S(r, t) = 21 B(r, t) + B(t, r) and A(r, t) = 21 B(r, t) − B(t, r) . Both are G0 -invariant, and consequently their degeneracy subspaces are invariant. Since the representation space R is irreducible, it follows that each is either nondegenerate or vanishes identically. But both being nondegenerate would violate the fact that up to a constant multiple there is only one equivariant isomorphism in End(R). Therefore B is either symmetric or alternating as claimed. Now let H := HomG0 (R, V ) be the space of G0 -equivariant linear mappings from R to V . Its dual space is H ∗ = HomG0 (R ∗ , V ∗ ). The key space for our first considerations is (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ). (Here, and throughout this paper, tensor products are understood to be tensor products over the field of complex numbers.) Note that G0 acts on it by g(h ⊗ r + f ⊗ t) = h ⊗ g(r) + f ⊗ g(t) . We can apply h ∈ H to r ∈ R to form h(r) ∈ V . Since h is G0 -equivariant we have g ·h(r) = h(g(r)). The same goes for the corresponding objects on the dual side. Thus in our finite-dimensional setting the following is immediate. (Once again, note the change of meaning of the symbol ε as compared to Sect. 2.1.) Proposition 3.3. If H = HomG0 (R, V ) and H ∗ = HomG0 (R ∗ , V ∗ ) the map ε : (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) → V ⊕ V ∗ = W , h ⊗ r + f ⊗ t → h(r) + f (t) , is a G0 -equivariant isomorphism. Transferring the unitary structure from W to (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) induces a unitary structure on H ⊕ H ∗ . For this, note for example that for h1 ⊗ r1 and h2 ⊗ r2 in H ⊗ R we have h1 ⊗ r1 , h2 ⊗ r2 H ⊗R := h1 (r1 ), h2 (r2 )V .
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Observe that for h1 and h2 fixed, the right-hand side of this equality defines a G0 -invariant unitary structure on R which is unique up to a multiplicative constant. Thus we define , H by h1 ⊗ r1 , h2 ⊗ r2 H ⊗R = h1 , h2 H · r1 , r2 R . Given the fixed choice of , R this definition is canonical. We will in fact transfer all of our considerations for V ⊕ V ∗ to the space H ⊕ H ∗ , the latter being equipped with the unitary structure defined as above. One of the key points for this is to understand how to express a G0 -invariant endomorphism S ∈ EndG0 (V ⊕ V ∗ ) ∼ =ε EndG0 (H ⊗ R ⊕ H ∗ ⊗ R ∗ ) as an element of End(H ⊕ H ∗ ). Also, we must understand the role of time reversal. In this regard the two cases λ = λ∗ and λ = λ∗ pose slightly different problems. Before going into these in the next sections, we note several facts which are independent of the case. First, let V1 and V2 be vector spaces where G0 acts trivially, and let R1 and R2 be arbitrary G0 -representation spaces. Proposition 3.4. HomG0 (V1 ⊗ R1 , V2 ⊗ R2 ) = Hom(V1 , V2 ) ⊗ HomG0 (R1 , R2 ) . Proof. Note that Hom(V1 ⊗ R1 , V2 ⊗ R2 ) = Hom(V1 , V2 ) ⊗ Hom(R1 , R2 ), and let (ϕ1 , . . . , ϕm ) be a basis of Hom(V1 , V2 ). Then for every element S of Hom(V1 , V2 ) ⊗ Hom(R1 , R2 ) there are unique elements ψ1 , . . . , ψm so that S = ϕi ⊗ ψi . If S is G0 -equivariant, then ϕi ⊗ (g ◦ ψi ◦ g −1 ) , S = g ◦ S ◦ g −1 = and the desired result follows from the uniqueness statement.
Our second general remark concerns the way in which a distinguished time-reversal symmetry T is transferred to an antilinear endomorphism of H ⊗R⊕H ∗ ⊗R ∗ . Let us consider for example the case of mixing where it is sufficient to understand T : H ⊗R → H ∗ ⊗R ∗ . For that purpose we view End(H ⊗ R) as End(H ) ⊗ End(R), let (ϕ1 , . . . , ϕm ) be a basis of End(H ) and write = CT = ϕi ⊗ ψi for ψ1 , . . . , ψm ∈ End(R). Now T is equivariant in the sense that T ◦ g = a(g) ◦ T , where a is the automorphism of G0 determined by conjugation with gT . Thus, since the C-antilinear operator C intertwines G0 -actions, the C-linear mapping = CT is invariant with respect to the twisted conjugation → a(g)g −1 . Consequently, every ψi is invariant with respect to this conjugation. This means that the ψi : R → R are equivariant with respect to the original G0 -representation on the domain space and the new G0 -action, v → a(g)(v), on the image space. But by Prop. 3.1, up to a constant multiple there is only one such element of End(R), i.e., we may assume that =ϕ⊗ψ , where ψ is unique up to a multiplicative constant.
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Note further that C is also of this factorized form. Indeed, we have h ⊗ r, ·H ⊗R = h, ·H r, ·R , and if γ : H → H ∗ is defined by h → h, ·H , then C = γ ⊗ ι. Furthermore, since and C are pure tensors, so is T = C = TH ⊗ TR , with the factors being antilinear mappings TH = γ ◦ ϕ : H → H ∗ and TR = ι ◦ ψ : R → R ∗ . Of course we have only considered a piece of T , and that only in the case of mixing. However, exactly the same arguments apply to the other piece and also in the case of nonmixing. Thus we have the following observation. Proposition 3.5. The induced map T : (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) → (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) , is the sum T = A1 ⊗ B1 + A2 ⊗ B2 of pure tensors. In the case of mixing this means that A1 ⊗ B1 is an antilinear mapping from H ⊗ R to H ∗ ⊗ R ∗ and vice versa for A2 ⊗ B2 . If T doesn’t mix, then A1 ⊗B1 : H ⊗R → H ⊗R and A2 ⊗B2 : H ∗ ⊗R ∗ → H ∗ ⊗R ∗ . In this case we impose the natural condition that the Ai and Bi be antiunitary. For later purposes we note that this condition determines the factors only up to multiplication by a complex number of unit modulus. Using the formula C = γ ⊗ ι and the fact that C commutes with T , one immediately computes A2 ⊗ B2 from A1 ⊗ B1 (or vice versa). The involutory property T 2 = ±Id also adds strong restrictions. Of course there may be two distinguished time reversals, T and T1 , and we require that they commute with C and T1 T = ±T T1 . These properties are automatically transferred at this level, because the transfer process from (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) to V ⊕ V ∗ is an isomorphism. Finally, we prove an identity which is essential for transferring the complex bilinear form. For this we begin with h ⊗ r + f ⊗ t ∈ (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) , apply ε to obtain h(r) + f (t), and then apply the linear function f (t) ∈ V ∗ to the vector h(r) ∈ V . The result f (t)(h(r)) is to be compared to the product f (h) t (r). Recall that the dimension of the vector space R is denoted by d. Proposition 3.6. f (t)(h(r)) = d −1 f (h) t (r) . Before beginning the proof, which uses bases for the various spaces, we set the notation and prove a preliminary lemma. Let m denote the multiplicity of the component V and fix an identification V ⊕ V ∗ = R ⊕ . . . ⊕ R ⊕ R∗ ⊕ . . . ⊕ R∗ with m summands of R and R ∗ . Let (e1 , . . . , ed ) be a basis of R and (ϑ1 , . . . , ϑd ) be its dual basis. These define bases (e1k , . . . , edk ) and (ϑ1k , . . . , ϑdk ) of the corresponding k th summands above. Let IRk and IRk ∗ be the respective identity mappings.
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P. Heinzner, A. Huckleberry, M.R. Zirnbauer
Lemma 3.7. IR ∗ (IRk ) = δk d . Proof. Expressing the operators in the bases, i.e., ϑik ⊗ eik and IR ∗ = ej ⊗ ϑj , IRk = i
one has IR∗ (IRk ) =
j
ϑik (ej ) ϑj (eik ) =
i,j
which is the statement of the lemma.
δ k i,j ij
= δk d ,
hk IRk , and f ∈ Proof of Prop. 3.6. We expand h ∈ H = HomG0 (R, V ) as h = ∗ ∗ ∗ H = HomG0 (R , V ) as f = f IR ∗ . If r = ri ei and t = tj ϑj , then h(r) = hk ri eik and f (t) = f tj ϑj . i,k
j,
Thus f (t)(h(r)) =
ij k
δijk f hk tj ri =
k
fk hk t (r) .
Proposition 3.6 now follows from the above lemma which implies that f (h) = d f k hk . 3.2. The case where λ = λ∗ . Recall that our goal is to canonically transfer the data on V ⊕ V ∗ to H ⊕ H ∗ , thus removing G0 from the picture. In the case where λ = λ∗ this is a particularly simple task. First, we apply Prop. 3.4 to transfer elements of EndG0 (V ⊕ V ∗ ). In the case at hand HomG0 (R, R ∗ ) and HomG0 (R ∗ , R) are both zero, and both EndG0 (R) and EndG0 (R ∗ ) are isomorphic to C. Thus it follows from Prop. 3.4 that EndG0 (V ⊕ V ∗ ) ∼ = EndG0 (H ⊗ R ⊕ H ∗ ⊗ R ∗ ) ∼ = End(H ) ⊕ End(H ∗ ) → End(H ⊕ H ∗ ) . We always normalize operators in EndG0 (H ⊗ R) to the form ϕ ⊗ IdR and normalize operators in EndG0 (H ∗ ⊗ R ∗ ) in a similar way. Thus we identify EndG0 (V ⊕ V ∗ ) with End(H ) ⊕ End(H ∗ ) as a subspace of End(H ⊕ H ∗ ) and have the following result. Proposition 3.8. The condition that an operator in EndG0 (V ⊕ V ∗ ) respects the unitary structure on V ⊕V ∗ is equivalent to the canonically transferred operator in End(H ⊕H ∗ ) respecting the canonically transferred unitary structure on H ⊕ H ∗ . Now let us turn to the condition of compatibility with a transferred time-reversal operator T : H ⊗ R ⊕ H ∗ ⊗ R ∗ → H ⊗ R ⊕ H ∗ ⊗ R ∗ . There are a number of cases, depending on whether or not T mixes and which of the conditions T 2 = −Id or T 2 = Id are satisfied. The arguments are essentially the same in every case. Let us first go through the details in one of them, the mixing case where T 2 = −Id. To be consistent with the
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slightly more complicated discussion in the case where λ = λ∗ , let us write this in matrix notation. For A ∈ End(H ) and D ∈ End(H ∗ ), we regard A ⊗ IdR 0 M= 0 D ⊗ IdR ∗ as the associated transformation in EndG0 (H ⊗R⊕H ∗ ⊗R ∗ ). To construct the transferred time-reversal operator recall the statement of Prop. 3.5. In the setting under consideration T squares to minus the identity; it is therefore expressed as 0 −α −1 ⊗ β −1 T = , α⊗β 0 where α : H → H ∗ and β : R → R ∗ are complex antilinear. Note that since α ⊗ β = z α ⊗ z−1 β, the mappings α and β are determined only up to a common multiplicative constant z ∈ C \ {0}. Conjugation of M in EndG0 (H ⊗ R ⊕ H ∗ ⊗ R ∗ ) by T yields −1 α D α ⊗ IdR 0 −1 T MT = . 0 αAα −1 ⊗ IdR ∗ Clearly, compatibility of M with T here means that D = αAα −1 . Formulating this in a less detailed way gives the appropriate statement: conjugation of M in EndG0 (H ⊗ R ⊕ H ∗ ⊗ R ∗ ) by T yields the same compatibility condition as conjugating A 0 0 ∓α −1 by . 0 D α 0 Here the sign in front of α −1 is arbitrary. For definiteness we choose it in such a way that the transferred time-reversal operator has the same involutory property T 2 = −Id or T 2 = Id as the original operator; in the case under consideration this means that we choose the minus sign. Proposition 3.9. There is a transferred time-reversal operator T : H ⊕ H ∗ → H ⊕ H ∗ which satisfies either T 2 = −Id or T 2 = Id. It mixes if and only if the original operator mixes, and a canonically transferred mapping in End(H ⊕ H ∗ ) commutes with it if and only if the original mapping in EndG0 (V ⊕V ∗ ) commutes with the original time-reversal operator. Proof. It only remains to handle the case of nonmixing, e.g., when T 2 = −Id. As we have seen, T : H ⊗ R → H ⊗ R is a pure tensor: T |H ⊗R = α ⊗ β , which gives T 2 |H ⊗R = α 2 ⊗ β 2 = −IdH ⊗ IdR in the case at hand. Since the induced map β : R → R is antiunitary by convention, we have β 2 = z × IdR with |z| = 1. Associativity (β 2 · β = β · β 2 ) then implies z = ±1. Unlike the case of mixing, β now plays a role through its parity. If β 2 = +IdR , the transferred time-reversal operator α on H still satisfies α 2 = −IdH . On the other hand, if β 2 = −IdR we have α 2 = +IdH instead. Thus the involutory property T 2 = ±Id is passed on to the transferred time-reversal operator, but depending on the involutory character of β the parity may change.
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P. Heinzner, A. Huckleberry, M.R. Zirnbauer
We remind the reader that two distinguished time-reversal symmetries may be present. The above shows that both can be transferred with appropriate involutory properties. Further, it must be shown that they can be transferred (along with C) so that T C = CT , T1 C = CT1 , and T1 T = ±T T1 still hold. Even if there is just one such operator, it must be shown that the transferred operator can be chosen to satisfy T C = CT . Since the discussion for this is the same as in the case where λ = λ∗ , we postpone it to Sect. 3.4. Finally, we turn to the problem of transferring the complex bilinear form on V ⊕ V ∗ to H ⊕ H ∗ . If b denotes the pullback by ε of the canonical symmetric bilinear form on V ⊕ V ∗ , then by Prop. 3.6, b(h1 ⊗ r1 + f1 ⊗ t1 , h2 ⊗ r2 + f2 ⊗ t2 ) = d −1 (f1 (h2 )t1 (r2 ) + f2 (h1 )t2 (r1 )) . Now in this case, i.e., where λ = λ∗ , the G0 -invariant endomorphisms are acting on A ⊗ IdR 0 , where H ⊗ R ⊕ H ∗ ⊗ R ∗ by 0 D ⊗ IdR ∗ A ⊕ D ∈ End(H ) ⊕ End(H ∗ ) → End(H ⊕ H ∗ ) . Inserting the operator A ⊕ D into the above expression for b we have the following fact involving the canonical symmetric bilinear form s on H ⊕ H ∗ , s(h1 + f1 , h2 + f2 ) = f1 (h2 ) + f2 (h1 ) . Proposition 3.10. A map in EndG0 (V ⊕ V ∗ ) respects the canonical symmetric bilinear form if and only if the transferred map in End(H )⊕End(H ∗ ) → End(H ⊕H ∗ ) respects the canonical symmetric bilinear form s on H ⊕ H ∗ . In summary, we have shown that if λ = λ∗ , then all relevant structures on V ⊕ V ∗ transfer to data of essentially the same type on H ⊕ H ∗ (the only exception being that the parity of the transferred time-reversal operator may be reversed). In this case EndG0 (V ⊕ V ∗ ) is canonically isomorphic to End(H ) ⊕ End(H ∗ ) → End(H ⊕ H ∗ ). An operator in EndG0 (V ⊕ V ∗ ) respects the original structures if and only if the corresponding operator in End(H ⊕ H ∗ ) respects the transferred structures on H ⊕ H ∗ . The latter are the transferred unitary structure, induced time reversal and the symmetric bilinear form s. 3.3. The case where λ = λ∗ . Throughout this section it is assumed that λ = λ∗ , and ψ : R → R ∗ is a G0 -equivariant isomorphism. Thus we have the identification ∼ H ⊗ R ⊕ H ∗ ⊗ R∗, H ⊗ R ⊕ H∗ ⊗ R = h ⊗ r + f ⊗ t → h ⊗ r + f ⊗ ψ(t) . Applying Prop. 3.4 to each component of an operator in EndG0 (H ⊗ R ⊕ H ∗ ⊗ R) it follows that EndG0 (H ⊗ R ⊕ H ∗ ⊗ R ∗ ) ∼ = End(H ⊕ H ∗ ) . We therefore identify End(H ⊕H ∗ ) with EndG0 (H ⊗R ⊕H ∗ ⊗R ∗ ) = EndG0 (V ⊕V ∗ ) by the mapping AB A ⊗ IdR B ⊗ ψ −1 M= → . CD C ⊗ ψ D ⊗ IdR ∗
Symmetry Classes of Disordered Fermions
747
Recall the induced unitary structure which is defined, e.g., on H ⊗ R by h1 ⊗ r1 , h2 ⊗ r2 H ⊗R := h1 (r1 ), h2 (r2 )V = h1 , h2 H r1 , r2 R . It is easy to verify that this defines a unitary structure on H ⊕ H ∗ with the desired property: a map in EndG0 (V ⊕ V ∗ ) preserves the given unitary structure on V ⊕ V ∗ if and only if the transferred map M preserves the induced unitary structure on H ⊕ H ∗ . Now let us consider time reversal. For example, take the case of nonmixing where T1 : H ⊗ R → H ⊗ R. Using Prop. 3.5 we have α⊗β 0 T1 = , 0 α˜ ⊗ β˜ A ⊗ IdR B ⊗ ψ −1 and conjugating the transformation at the level of operators on C ⊗ ψ D ⊗ IdR ∗ ∗ ∗ H ⊗ R ⊕ H ⊗ R yields αAα −1 ⊗ IdR αBα˜ −1 ⊗ βψ −1 β˜ −1 . ˜ −1 αD αCα ˜ −1 ⊗ βψβ ˜ α˜ −1 ⊗ IdR ∗ Now, as has been mentioned in Sect. 3.1, the equivariant antiunitary maps β and β˜ are only unique up to multiplicative constants of unit modulus. They will be chosen in the next subsection so that the distinguished time-reversal operator(s) and the unitary structure C commute. These choices having been made, we make a compatible choice ˜ −1 = ψ. In this way, in the case where T1 is nonmixing as above, of ψ so that βψβ conjugation of the matrix M by T1 is given by AB αAα −1 αBα˜ −1 → . (3.1) CD αCα ˜ −1 αD ˜ α˜ −1 Thus the transferred time-reversal operator is simply given by T1 = α ⊕ α˜ on H ⊕ H ∗ . Consider now the case of a mixing time-reversal symmetry T where 0 α −1 ⊗ β −1 T = εT α ⊗ β 0 with εT = ±1. In this case the compatibility condition on ψ is βψ −1 β = εβ ψ, with εβ = ±1. If this holds, conjugation of M by T is given by −1 AB α Dα εα α −1 Cα −1 → (3.2) CD εα αBα αAα −1 with εα = εβ εT . In this case the appropriate transferred operator is given by 0 α −1 T = . εα α 0 Given the (essentially unique) choices of the tensor-product representations of T , T1 and C which are defined by T1 T = ±T T1 and by the conditions that T and T1 commute with C, we show in Sect. 3.4 that there is a unique choice of ψ so that both of these compatibility conditions (from T1 and T ) on ψ hold.
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If we are in the nonmixing case β : R → R, and it so happens that β is G0 -invariant, then the two alternatives for the involutory property of T (actually, T1 ) can be distinguished by the type of the unitary representation R as follows. Defining ι : R → R ∗ by r → r, ·R as before, consider the unitary mapping ψ : R → R ∗ given as the composition ψ = ι ◦ β. Since β is G0 -invariant, ψ is G0 -equivariant, and the statement of Lemma 3.2 applies. Using the antiunitarity of β one has ψ(r)(t) = βr, tR = β 2 r, βtR = ψ(t)(β 2 r) , and therefore the following statement is immediate. Lemma 3.11. The parity of an antiunitary and G0 -invariant mapping β : R → R is determined by the parity of the irreducible G0 -representation space R; i.e., β satisfies β 2 = IdR resp. β 2 = −IdR if R carries an invariant C-bilinear form which is symmetric resp. alternating. If β 2 = IdR , the transferred time reversal satisfies T 2 = −Id or T 2 = Id if the original time reversal has these properties. On the other hand, if β 2 = −IdR , then the properties are reversed; e.g., if T 2 = −Id on the original space, then transferred time reversal satisfies T 2 = Id. We again remind the reader that we must check that the transferred time-reversal operator(s) and C can be chosen compatibly. It turns out that there is in fact just enough freedom in the choice of the constants to achieve this (see Sect. 3.4). Example. An example of particular importance in physics is the transfer of the (true) time reversal T in the case where all spin rotations are symmetries. On fundamental grounds, T is a (nonmixing) operator which commutes with the spin-rotation group SU2 and satisfies T 2 = (−1)n Id on quantum mechanical states with spin S = n/2. Let V = H ⊗ Cn+1 be the tensor product of a vector space H with the spin n/2 representation space of SU2 . For simplicity assume that there are no further symmetries. Our Nambu space is already in the form V ⊕ V ∗ = (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ). Thus the reduced space is H ⊕ H ∗ . Let the time-reversal operator on V = H ⊗ Cn+1 be written T = α ⊗ β. The SU2 -representation space Cn+1 is known to have parity +1 (symmetric invariant form) for n even, and −1 (alternating invariant form) for n odd. By Lemma 3.11 this implies β 2 = (−1)n Id. The situation on the dual space V ∗ is the same. Thus in this case, since T 2 = (−1)n Id, the transferred time-reversal operator α : H ⊕ H ∗ → H ⊕ H ∗ always satisfies α 2 = +IdH ⊕H ∗ , independent of the spin. Now let us turn to the problem of transferring the complex bilinear form. For this Lemma 3.2 is an essential fact. Earlier we identified H ⊗ R ⊕ H ∗ ⊗ R ∗ with V ⊕ V ∗ by the map ε : h ⊗ r + f ⊗ t → h(r) + f (t). Using this along with Prop. 3.6 we now transfer the canonical symmetric bilinear form on V ⊕ V ∗ to H ⊕ H ∗ . For this let s (resp. a) denote the canonical symmetric (resp. alternating) form on H ⊕ H ∗ . Proposition 3.12. Depending on ψ being symmetric or alternating, a transferred map in End(H ⊕ H ∗ ) respects the canonical symmetric form s or alternating form a if and only if the original endomorphism in EndG0 (V ⊕ V ∗ ) respects the canonical symmetric complex bilinear form on V ⊕ V ∗ . Proof. We give the proof for the case where ψ is alternating. The proof in the symmetric case is completely analogous.
Symmetry Classes of Disordered Fermions
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Let M =
AB ∈ End(H ⊕ H ∗ ) act as a G0 -invariant operator CD A ⊗ IdR B ⊗ ψ −1 C ⊗ ψ D ⊗ IdR ∗
on H ⊗ R ⊕ H ∗ ⊗ R ∗ and let b be the symmetric complex bilinear form on this space which is induced from the canonical symmetric form on V ⊕ V ∗ . We assume that M ∈ GL(H ⊕ H ∗ ) and give the proof in terms of the isometry property b(Mv, Mw) = b(v, w). Let us do this in a series of cases. First, for h1 ⊗ r1 and h2 ⊗ r2 in H ⊗ R, b(M(h1 ⊗ r1 ), M(h2 ⊗ r2 )) = b(Ah1 ⊗ r1 + Ch1 ⊗ ψ(r1 ), Ah2 ⊗ r2 + Ch2 ⊗ ψ(r2 )) = Ch1 (Ah2 ) ψ(r1 )(r2 )/d + Ch2 (Ah1 ) ψ(r2 )(r1 )/d = a(Ah1 + Ch1 , Ah2 + Ch2 ) ψ(r1 )(r2 )/d . When M is the identity this becomes b(h1 ⊗ r1 , h2 ⊗ r2 ) = a(h1 , h2 ) ψ(r1 )(r2 )/d . Therefore b(h1 ⊗ r1 , h2 ⊗ r2 ) = b(M(h1 ⊗ r1 ), M(h2 ⊗ r2 )) if and only if a(h1 , h2 ) = a(M(h1 ), M(h2 )). For f1 ⊗ t1 , f2 ⊗ t2 ∈ H ∗ ⊗ R ∗ the discussion is analogous. For h ⊗ r ∈ H ⊗ R and f ⊗ t ∈ H ∗ ⊗ R ∗ we have a similar calculation: b(M(f ⊗ t), M(h ⊗ r)) = b(Bf ⊗ ψ −1 (t) + Df ⊗ t, Ah ⊗ r + Ch ⊗ ψ(r)) = Df (Ah) t (r)/d + Ch(Bf ) ψ(r)(ψ −1 (t))/d = a(M(f ), M(h)) t (r)/d . Of course the analogous identity holds for b(M(h ⊗ r), M(f ⊗ t)).
Remark. To avoid making sign errors and misidentifications in later computations, we find it helpful to transfer the particle-hole conjugation operator C along with the complex bilinear form. This is done by insisting that the statement of Lemma 2.2 remains true after the transfer. Thus the relation b(Cw1 , w2 ) = w1 , w2 continues to hold in all cases. By an almost identical variant of the computation that led to Lemma 3.11, the transferred operator C has parity C 2 = +Id or C 2 = −Id depending on whether the transferred bilinear form is symmetric or alternating. 3.4. Precise choice of time-reversal transfer. Recalling the situation of this section, we have assumed that the distinguished time-reversal operator(s) stabilize the initial block V ⊕ V ∗ , and we have transferred all structures to the space (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ) which is isomorphic to V ⊕ V ∗ . The time-reversal operator(s) T and the operator C are given by (2 × 2)-matrices of pure tensors on this space. The space of endomorphisms B that commute with the G0 action is identified with End(H ⊕ H ∗ ) or End(H ) ⊕ End(H ∗ ) depending on whether or not λ = λ∗ . The good Hamiltonians B anticommute with C, and commute with the time-reversal operator(s) T . If the matrix of pure tensors representing the antiunitary operator C (resp. T ) has entries γ ⊗δ, this means that B anticommutes (resp. commutes)
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P. Heinzner, A. Huckleberry, M.R. Zirnbauer
with the matrices defined by the operators γ . Although the pure tensor decomposition is not unique, this statement is independent of that decomposition. It has been shown above that the transferred operators T1 , T , and C on H ⊕ H ∗ , i.e., those defined by the operators γ , can be chosen with the desired involutory properties. It will now be shown that there is just enough freedom to insure that T C = CT ,
T1 C = CT1 ,
T1 T = ±T T1 ,
still hold after transferral. After these conditions have been met, we show as promised that ψ : R → R ∗ can be chosen in a unique way so that the compatibility conditions of Sect. 3.3 hold, i.e., so that it makes sense to define the transferred operators by the first factors of the tensor-product representations. We carry this out in the case where λ = λ∗ and two distinguished time-reversal operators are present. All other cases are either subcases of this or are much simpler. The operator C always mixes. We will always choose it to be of the form C = γ ⊗ ι : H ⊗ R → H ∗ ⊗ R ∗ and C = γ −1 ⊗ ι−1 : H ∗ ⊗ R ∗ → H ⊗ R. Of course this is in the case where b is symmetric. If b is alternating, then we have C 2 = −Id, and we make the necessary sign change. Here we restrict to the case where T 2 = T12 = Id. The various other involutory properties make no difference in the argument. Just as in the case of C we choose T = α ⊗ β : H ⊗ R → H ∗ ⊗ R ∗ and T = α −1 ⊗ β −1 : H ∗ ⊗ R ∗ → H ⊗ R. Similarly, we choose T1 = α1 ⊗ β1 : H ⊗ R → H ⊗ R and T1 = α2 ⊗ β2 : H ∗ ⊗ R ∗ → H ∗ ⊗ R ∗ . On (H ⊗ R) ⊕ (H ∗ ⊗ R ∗ ), the operators T and T1 commute with C, and we have T1 T = ±T T1 . We now choose the tensor representations so that the same relations hold for the induced operators on the first factors. If α, α1 , α2 , and γ are any choices for the first factors of the tensor-product representations of T , T1 and C, then there exist constants c1 , c2 and c3 so that α2 α = c1 αα1 (from T T1 = ±T1 T ), γ α1 = c2 α2 γ (from CT1 = T1 C), and γ α −1 γ = c3 α (CT = T C). Let α˜ = ξ α, γ˜ = ηγ , and α˜ i = zi α (for i = 1, 2), where ξ , η and zi are complex numbers yet to be determined. Just as the ci , these constants are of modulus one. The scaled operators satisfy α˜ 2 α˜ = χ1 c1 α˜ α˜ 1 , γ˜ α˜ 1 = χ2 c2 α˜ 2 γ˜ , and γ˜ α˜ −1 γ˜ = χ3 c3 α, ˜ where χ1 = ξ −2 z1 z2 , χ2 = η2 (z1 z2 )−1 , and χ3 = ξ −2 η2 . Observe that the characters χi satisfy the relation χ1 χ2 = χ3 , and that, e.g., χ2 and χ3 are independent. The constants ci satisfy an analogous relation. For this first use γ α1 γ −1 = c2 α2 and γ α −1 γ = c3 α to obtain γ α1 α −1 γ = (c2 /c3 )α2 α. Then compose both sides of this equation with the inverse of α1 on the right and use the relation α2 αα1−1 = c1 α to obtain γ α1 α −1 γ α1−1 = (c1 c2 /c3 )α. Now γ α1−1 = (c2 α2 )−1 γ . Thus γ α1 α −1 γ α1−1 = γ α1 α −1 α2−1 c2−1 γ = c2−1 γ c1 α −1 γ = (c1 c2 )−1 c3 α , and hence c1 c2 /c3 = c3 /c1 c2 , i.e., c12 c22 = c32 . Since χ2 and χ3 are independent, we can choose the scaling numbers so that c2 = c3 = 1, thereby arranging that CT = T C and CT1 = T1 C still hold after transferral. To preserve these relations we must now keep χ2 and χ3 fixed at unity, which from χ1 χ2 = χ3 implies that χ1 = 1. Since c32 = c12 c22 , we then conclude that c1 takes one of the two values ±1, and further scaling does not change this constant. In summary we have the following result. Proposition 3.13. The transferred operators T1 , T and C can be chosen so that T1 C = CT1 , T C = CT , and T1 T = ±T T1 . Assuming that the time-reversal operators have
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been transferred to commute with C in this way, the relation T1 T = ±T T1 is automatic and further scaling does not change the sign. Furthermore, the C-linear isomorphism ψ : R → R ∗ can be chosen to meet the compatibility conditions which determine the conjugation rules (3.1) and (3.2). Proof. It remains to prove that ψ can be chosen as stated. For the nonmixing operator T1 the compatibility condition is β2 ψβ1−1 = ψ. Given some choice of ψ (which we will modify) there is a constant c ∈ C so that β2 ψβ1−1 = cψ. This constant c is unimodular since β1 and β2 are antiunitary. To satisfy the compatibility condition, replace ψ by ξ ψ, where ξ¯ ξ −1 c = 1. Note that this choice of ξ only determines its argument. Turning to the compatibility condition βψ −1 β = εβ ψ for the mixing operator T , we start from cψ = βψ −1 β for some other c ∈ C, and use the C-antilinearity of β to deduce ψ −1 = c¯ β −1 ψβ −1 . Multiplying expressions gives c = c¯ ∈ R. Then, rescaling ψ to ξ ψ, the compatibility condition is achieved by setting εβ := c/|c| and solving |ξ |2 = |c|. Since this rescaling (with ξ ∈ R) does not affect the compatibility condition for the nonmixing operator, we have determined the desired isomorphism ψ. Finally, since C is a pure tensor, it follows from our representation of the transferred bilinear form b that cb(Ch1 , h2 ) = h1 , h2 for some constant c. Thus we replace b by cb and obtain the following final transferred setup on H ⊕ H ∗ : • The canonical bilinear form b which is either symmetric or alternating. • A unitary structure , which is compatible with b in the sense that b(Ch1 , h2 ) = h1 , h2 . The operator C : H ↔ H ∗ satisfies either C 2 = Id or C 2 = −Id, depending on b being symmetric or alternating. • Either zero, one, or two time-reversal operators. They are antiunitary and commute with C. In the case of two, T is mixing and T1 is nonmixing. In the case of one, both mixing and nonmixing are allowed. The same involutory properties hold as before transfer, but signs might change, i.e., if T 2 = Id holds before transfer, then it is possible that T 2 = −Id afterwards. Furthermore, T1 T = ±T T1 , and consequently the unitary product P := T T1 satisfies P 2 = ±Id. In the following sections all of the symmetric spaces which occur in our basic model will be described, using the transferred setup. This means that we describe the subspace of Hermitian operators in End(H ) ⊕ End(H ∗ ) or End(H ⊕ H ∗ ) which are compatible with b and the T -symmetries. We first handle the case of one or no time-reversal operator (Sect. 4), and then carry out the classification when both T and T1 are present (Sect. 5). The final classification result, Theorem 1.1, then follows. 4. Classification: At Most One Distinguished Time Reversal This section is devoted to giving a precise statement of Theorem 1.1 and its proof in the case where at most one distinguished time-reversal symmetry is present. Combining this with the results of Sect. 3, we obtain a precise description of the blocks that occur in the model motivated and described in Sects. 1 and 2. 4.1. Statement of the main result. Throughout this section, V denotes a finitedimensional unitary vector space. The associated space W = V ⊕ V ∗ is equipped with the canonically induced unitary structure , and C-antilinear map C : V → V ∗ ,
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v → v, ·. The results of the previous section allow us to completely eliminate G0 from the discussion so that it is only necessary to consider the following data: • The relevant space E of endomorphisms. This is either the full space End(W ) or End(V ) ⊕ End(V ∗ ) embedded as usual in End(W ). • The canonical complex bilinear form b : W × W → C. This is either the symmetric form s which is given by s(v1 + f1 , v2 + f2 ) = f1 (v2 ) + f2 (v1 ) , or the alternating form a which is given by a(v1 + f1 , v2 + f2 ) = f1 (v2 ) − f2 (v1 ) . Equivalently, C : V → V ∗ is extended to a C-antilinear mapping C : W → W by C 2 = +Id resp. C 2 = −Id, and b(Cw1 , w2 ) = w1 , w2 holds in all cases. • The antiunitary mapping T : W → W , which satisfies either T 2 = −Id or T 2 = Id. We say that T is nonmixing if T |V : V → V and T |V ∗ : V ∗ → V ∗ . If T |V : V → V ∗ , then we refer to T as mixing. In all cases T commutes with C. We also include the case where T is not present. Fixing one of these properties each, we refer to (V , E, b, T ) as block data; e.g., E = End(W ), b = s, T 2 = −Id and T being nonmixing would be such a choice. Our main result describes the symmetric spaces associated to given block data. Let us state this at the Lie algebra level, where for convenience of formulation we only consider the case of trace-free operators. In order to state this result, it is necessary to introduce some notation. Given block data (V , E, b, T ), let g be the subspace of E of antihermitian operators A which are compatible with b in the sense that b(Aw1 , w2 ) + b(w1 , Aw2 ) = 0 for all w1 , w2 ∈ W . It will be shown that g is a Lie subalgebra of E which is invariant under conjugation A → T AT −1 with T . This defines a Lie algebra automorphism θ :g→g,
A → T AT −1 ,
which is usually called a Cartan involution. If k := Fix(θ ) = {A ∈ g : θ (A) = A} and p is the space {A ∈ g : θ (A) = −A} of antifixed points, then g=k⊕p is called the associated Cartan decomposition. The space H = H(V , E, b, T ) of Hermitian operators which are compatible with the block data is ip, which is identified with the infinitesimal version p = g/k. In order to give a smooth statement of our classification result, we recall that the Lie algebras sun , usp2n , and so2n are commonly referred to as being of type A, C, and D, respectively. By an irreducible ACD-symmetric space of compact type one means an (irreducible) compact symmetric space of any of these Lie algebras. With a slight exaggeration we use the same terminology in Theorem 4.1 below. The exaggeration is that the case so2n /(sop ⊕soq ) with p and q odd must be excluded in order for that theorem to be true. For the overall statement of Theorem 1.1 there is no danger of misinterpretation, as the case where p and q are odd does occur in the situation where two distinguished time-reversal symmetries are present (see Sect. 5).
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Theorem 4.1. Given block data (V , E, b, T ), the space H = H(V , E, b, T ) ∼ = g/k is the infinitesimal version of an irreducible ACD-symmetric space of compact type. Conversely, the infinitesimal version of any irreducible ACD-symmetric space of compact type can be constructed in this way. There are several remarks which should be made concerning this statement. First, as we have already noted, in order to give a smooth formulation, we have reduced to trace-free operators. As will be seen in the proof, there are several cases where without this assumption g would have a one-dimensional center. Secondly, recall that one of the important cases of a compact symmetric space is that of a compact Lie group K with the geodesic inversion symmetry at the identity being defined by k → k −1 . Usually one equips K with the action of G = K × K defined by left- and right-multiplication, and views the symmetric space as G/K, where the isotropy group K is diagonally embedded in G. The infinitesimal version is then (k ⊕ k)/k, and the automorphism θ : g → g is defined by (X1 , X2 ) → (X2 , X1 ). In this setting one speaks of symmetric spaces of type II. In our case the classical compact Lie algebras do indeed arise from appropriate block data, but in the situation where T does not leave the original space W invariant. In that setting, T maps W = W1 = V1 ⊕ V1∗ to W2 = V2 ⊕ V2∗ , which has different G0 representations from those in W1 . Thus the relevant block is W1 ⊕ W2 . Using the results of the previous section, in this case we also remove G0 from the picture. Nevertheless, we are left with a situation where the block is W1 ⊕ W2 and T : W1 → W2 . Thus we wish to allow situations of this type, i.e., where V ⊕ V ∗ is not T -invariant, to be allowed block data. These cases are treated separately in Sect. 4.4. The case where the symmetric space is just the compact group associated to g also arises when T is not present, i.e., when there is no condition which creates isotropy. Finally, as has already been indicated in Sect. 1, the appropriate homogeneous space version of Theorem 4.1 is given by replacing the infinitesimal symmetric space g/k by the Cartan-embedded symmetric space M ∼ = G/K. Here G is the simply connected group associated to g, a mapping θ : G → G is defined as the Lie group automorphism whose derivative at the identity is the Cartan involution of the Lie algebra, and M is the orbit of e ∈ G of the twisted G-action given by x → gx θ (g)−1 . 4.2. The associated symmetric space. In this and the next subsection we work in the context of simple block data (V , E, b, T ), where W = V ⊕ V ∗ is T -invariant. In the present subsection we prove the first half of Theorem 4.1, namely that H ∼ = g/k is an infinitesimal version of a classical symmetric space of compact type. This essentially amounts to showing that all the involutions which are involved commute. Let σ : E → E be the C-antilinear Lie-algebra involution that fixes the Lie algebra of the unitary group in E. If the adjoint operation A → A∗ is defined by Aw1 , w2 = w1 , A∗ w2 , then σ (A) = −A∗ . The transformations S ∈ E which are isometries of the canonical bilinear form satisfy b(Sw1 , Sw2 ) = b(w1 , w2 ) for all w1 , w2 ∈ W . Thus the appropriate Lie algebra involution is the C-linear automorphism τ :E→E,
A → −At ,
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where A → At is the adjoint operation defined by b(Aw1 , w2 ) = b(w1 , At w2 ) . Finally, let θ : E → E be the C-antilinear map defined by A → T AT −1 . Proposition 4.2. The operations A → A∗ and A → At are related by A∗ = CAt C −1 . Proof. From b(Cw1 , w2 ) = w1 , w2 and the definition of A → A∗ we have b(Aw1 , w2 ) = C −1 Aw1 , w2 = C −1 w1 , (C −1 AC)∗ w2 = b(w1 , C −1 A∗ Cw2 ) , i.e., At = C −1 A∗ C, independent of the case b = s or b = a.
Proposition 4.3. The involutions σ , τ and θ commute. Proof. Using C −1 A∗ Cw1 , w2 = w1 , C −1 ACw2 along with At = C −1 A∗ C we have (At )∗ = C −1 AC = (A∗ )t , and consequently σ τ = τ σ . Since T is antiunitary, one immediately shows from the definition of A∗ that w1 , T A∗ T −1 w2 = T AT −1 w1 , w2 . In other words, θ (σ (A)) = −T A∗ T −1 = −(T AT −1 )∗ = σ (θ (A)) . Finally, since θ (A) = T AT −1 and T commutes with C, it follows that θτ = τ θ .
Let s := Fix(τ ). Since θ and σ commute with τ , it follows that they restrict to Cantilinear involutions of the complex Lie algebra s. We denote these restrictions by the same letters. For future reference let us summarize the relevant formulas. Proposition 4.4. For A ∈ s it follows that σ (A) = CAC −1 and θ (A) = T AT −1 . The parity of C is C 2 = +Id for b = s symmetric, and C 2 = −Id for b = a alternating. The space g of antihermitian operators in E that respect b is therefore the Lie algebra of σ -fixed points in s. Since σ defines the unitary Lie algebra in E, it follows that g is a compact real form of s. Let us explicitly describe s and g. If E = End(W ) and b = s is symmetric, then s is the complex orthogonal Lie algebra so(W, s) ∼ = so2n (C). If E = End(W ) and b = a is alternating, then s is the complex symplectic Lie algebra sp(W, a) ∼ = sp2n (C). If E = End(V ) ⊕ End(V ∗ ), then in both cases for b it follows that its isometry group is SLC (V ) acting diagonally by its defining representation on V and its dual representation on V ∗ . In this case we have s = sl(V ) ∼ = sln (C). Note that this is a situation where we have used the trace-free condition to eliminate the one-dimensional center. For the discussion of g it is important to note that since σ (A) = CAC −1 , it follows that g just consists of the elements of s which commute with C.
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In the symmetric case b = s, where C defines a real structure on W , it is appropriate to consider the set of real points WR = {v + Cv : v ∈ V }. Thinking in terms of isometries, we regard G = exp(g) as being the group of R-linear isometries of the restriction of b = s to WR which are extended complex linearly to W . Note that in this case b|WR = 2 Re , , and that every R-linear transformation of WR which preserves Re , extends C-linearly to a unitary transformation of W . Thus, if E = End(W ) and b = s, then g is naturally identified with so(WR , s|WR ) ∼ = so2n (R). In the alternating case b = a, if E = End(W ), then as in the previous case, since σ defines u(W ) ⊂ E, it follows that its set g of fixed points in s is a compact real form of s. Since s is the complex symplectic Lie algebra sp(W, a) ∼ = sp2n (C), it follows that g is isomorphic to the Lie algebra usp2n of the unitary symplectic group. It is perhaps worth mentioning that C for b = a defines a quaternionic structure on the complex vector space W . Thus the condition A = CAC −1 defines the subalgebra gln (H) in End(W ). The further condition A = −A∗ shows that g can be identified with the algebra of quaternionic isometries, another way of seeing that g ∼ = usp2n . Finally, in the case where E = End(V ) ⊕ End(V ∗ ) we have already noted that s = sl(V ) which is acting diagonally. It is then immediate that in both the symmetric and alternating cases g = su(V ) ∼ = sun . Of course g acts diagonally as well. Let us summarize these results. Proposition 4.5. In the case where E = End(W ) the following hold: • If b = s is symmetric, then g ∼ = so2n (R). • If b = a is alternating, then g ∼ = usp2n . If E = End(V ) ⊕ End(V ∗ ), then g is isomorphic to sun and acts diagonally. Since θ commutes with σ , it stabilizes g. Hence, θ |g is a Cartan involution which defines a Cartan decomposition g=k⊕p of g into its (±1)-eigenspaces. The fixed subspace k = {A ∈ g : θ (A) = A} is a subalgebra and g/k is the infinitesimal version of a symmetric space of compact type. Recall that, given block data (V , E, b, T ), the associated space H = H(V , E, b, T ) ∼ = ip of structure-preserving Hamiltonians has been identified with p = g/k. Thus we have proved the first part of Theorem 4.1. The second part is proved in the next section by going through the possibilities in Prop. 4.5 along with the various possibilities for T . It should be noted that if T = ±C, then g = k, i.e., the symmetric space is just a point. Such a degenerate situation, where the set of Hamiltonians is trivial (consisting only of the zero Hamiltonian), never occurs in a well-posed physics setting. 4.3. Concrete description: symmetric spaces of type I. Here we describe the possibilities for each set of block data (V , E, b, T ) under the assumption that W = V ⊕ V ∗ is T invariant. The results are stated in terms of the ACD-symmetric spaces, with n := dimC V . The methods of proof of showing which symmetric spaces arise also show how to explicitly construct them. In the present subsection, all of these are compact irreducible classical symmetric spaces of type I in the notation of [H].
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4.3.1. The case E = End(V ) ⊕ End(V ∗ ) Under the assumption E = End(V ) ⊕ End(V ∗ ) it follows that g is just the unitary Lie algebra su(V ) ∼ = sun which is acting diagonally on W = V ⊕ V ∗ . This is independent of b being symmetric or alternating. Thus we need only consider the various possibilities for T . If T is not present, the symmetric space is g = sun . 1. T 2 = −Id, nonmixing: sun /satisfies uspn . Since T is nonmixing and satifies T 2 = −Id, it follows that T v1 , v2 = a(v1 , v2 ) is a C-linear symplectic structure on V which is compatible with v1 , v2 . Thus the dimension n of V must be even here. The facts that g is acting diagonally as su(V ) and that the elements of k are precisely those which commute with T , imply that k = uspn as announced. 2. T 2 = Id, nonmixing: sun /son . Since T and g are acting diagonally, as in the previous case it is enough to only discuss the matter on V . In this case T defines a real structure on V with VR = {v + T v : v ∈ V }, and the unitary isometries which commute with T are just those transformations which stabilize VR and preserve the restriction of , . Since x, yVR = Re x, yV for x, y ∈ VR , it follows that k = so(VR ) ∼ = son (R). 3. T 2 = ±Id, mixing: sun /s(up ⊕ uq ). Here it is convenient to introduce the unitary operator P = CT , which satisfies P 2 = Id or P 2 = −Id, depending on the parity of T . Denote the eigenvalues of P by u and −u. Since P does not mix, the condition that a diagonally acting unitary operator commutes with T (or equivalently, with P ) is just that it preserves the P eigenspace decomposition V = Vu ⊕ V−u . Since the two eigenspaces Vu and V−u are , -orthogonal, we have k = s (u(Vu ) ⊕ u(V−u )), and the desired result follows with p = dim Vu and q = dim V−u . In the case P 2 = −Id, if there existed a subspace VR of real points that was stabilized by P , then P would be a complex structure of VR and the dimensions of Vu and V−u would have to be equal. In general, however, no such space VR exists and the dimensions p and q are arbitrary. 4.3.2. The case E = End(W ), b = s In this case we have the advantage that we may restrict the entire discussion to the set of real points WR = Fix(C) = {v + Cv : v ∈ V } . Thus k is translated to being the Lie algebra of the group of isometries of 2 Re , on V . Here the Lie algebra g is so(WR ). Thus in the case where T is not present, the symmetric space is so2n (R). 1. T 2 = −Id, nonmixing or mixing: so2n (R)/un . Independent of whether or not it mixes, T |WR : WR → WR is a complex structure on WR . A transformation in SO(WR ) commutes with T if and only if it is holomorphic. Since Re , is T -invariant, this condition defines the unitary subalgebra k ∼ = un in g∼ = so2n (R). 2. T 2 = Id, nonmixing: so2n (R)/(son (R) ⊕ son (R)). Since T |WR : WR → WR , we have the decomposition WR = WR+ ⊕ WR− into the (±1)-eigenspaces of T . We still identify g with the Lie algebra of the group of isometries of WR equipped with the restricted form Re , . The subalgebra k, which is
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fixed by θ : X → T XT −1 , is the stabilizer so(WR+ ) ⊕ so(WR− ) of the above decomposition. Now let us compute the dimensions of the eigenspaces. In the case at hand T defines a real structure on both V and V ∗ . Since C commutes with T , it follows that Fix(T ) = VR ⊕ VR∗ is C-invariant. Thus WR+ = {v + Cv : v ∈ VR }. A similar argument shows that WR− = {v + Cv : v ∈ iVR }. 3. T 2 = Id, mixing: so2n (R)/(so2p (R) ⊕ so2q (R)). The exact same argument as above shows that k = so(WR+ )⊕so(WR− ). It only remains to show that the eigenspaces are even-dimensional. For this we consider the unitary operator P = CT which leaves both V and V ∗ invariant. Its (+1)-eigenspace W+1 is just the complexification of WR+ . The intersections of W+1 with V and V ∗ are interchanged by C, and therefore dimC W+1 =: 2p is even. Of course the same argument holds for W−1 . 4.3.3. The case E = End(W ), b = a Since in this case g is the Lie algebra of antihermitian endomorphisms which respect the alternating form a on W , it follows that g∼ = usp2n . Thus if T is not present the associated symmetric space is usp2n . If T is present, we let P := CT . The unitary operator P always commutes with T , and from a(w1 , w2 ) = C −1 w1 , w2 one infers that a(P w1 , P w2 ) = a(w1 , w2 ) in all cases, independent of T being mixing or not. The classification spelled out below follows from the fact that commutation with T is equivalent to preservation of the P -eigenspace decomposition of W . 1. T 2 = −Id, nonmixing: usp2n /(uspn ⊕ uspn ). In this case P 2 = Id, and T 2 = −Id forces n to be even. Let W be decomposed into P -eigenspaces as W = W+1 ⊕ W−1 . If w1 ∈ W+1 and w2 ∈ W−1 , then a(w1 , w2 ) = a(P w1 , P w2 ) = −a(w1 , w2 ) = 0 , and we see that W+1 and W−1 are a-orthogonal. The mixing operator P is traceless. Therefore the dimensions of W+1 and W−1 are equal, and both of them are symplectic subspaces of W . The fact that the decomposition W = W+1 ⊕ W−1 is also , -orthogonal therefore implies that k = usp(W+1 ) ⊕ usp(W−1 ). 2. T 2 = −Id, mixing: usp2n /(usp2p ⊕ usp2q ). Here, using the same argument as in the previous case, one shows that the P-eigenspace decomposition W = W+1 ⊕ W−1 still is a direct sum of a-orthogonal, complex symplectic subspaces. Since these are also , -orthogonal, it follows that k = usp(W+1 ) ⊕ usp(W−1 ). Note that in the present case the nonmixing operator P stabilizes the decomposition W = V ⊕ V ∗ . Thus, since P commutes with C, it ∗ and W ∗ follows that W+1 = V+1 ⊕ V+1 −1 = V−1 ⊕ V−1 . 2 3. T = Id, mixing or nonmixing: usp2n /un . In this case P 2 = −Id. Here a(P w1 , P w2 ) = a(w1 , w2 ) implies that the P-eigenspace decomposition W = W+i ⊕W−i is Lagrangian. (This means in particular dim W+i = dim W−i .) Thus its stabilizer in sp(W ) is the diagonally acting gl(W+i ). Since the decomposition is , -orthogonal, it follows that k = u(W+i ) ∼ = un . 4.4. Concrete description: symmetric spaces of type II. Recall the original situation where the symmetry group G0 is still in the picture. As described in Sect. 1 we select from the given Hilbert space a basic finite-dimensional G0 -invariant subspace V which
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is composed of irreducible subrepresentations all of which are equivalent to a fixed irreducible representation R. Although the initial block of interest is W = V ⊕ V ∗ , it is possible that it is not T -invariant and that it must be expanded. Let us formalize this situation by denoting the initial block by W1 = V1 ⊕ V1∗ . We then let P = CT and regard this as a unitary isomorphism P : W1 → W2 , where W2 = V2 ⊕ V2∗ is another initial block. For i ∈ {1, 2}, let Ri be the irreducible G0 -representation on Vi which induces the representation on Wi . The map P is equivariant, but only with respect to the automorphism a of G0 which is defined by gT -conjugation: P ◦ g = a(g) ◦ P . As a brief interlude, let us investigate the consequences of this automorphism a being inner versus outer. If a is inner there exists A ∈ G0 such that a(g) = A−1 gA and hence AP ◦g = g ◦AP for all g ∈ G0 . Thus AP : W1 → W2 is a G0 -equivariant isomorphism and we have either R1 ∼ = R2 or R1 ∼ = R2∗ depending on whether T is mixing or not. In either case we may build a new block W = V ⊕ V ∗ which is T -invariant so that the results of the previous section can be applied: if R1 ∼ = R2 , then we let V := V1 ⊕ V2 and if R1 ∼ = R2∗ , then V := V1 ⊕ V2∗ . If the G0 -automorphism g → a(g) is outer, it may still happen that R1 ∼ = R2 or R1 ∼ = R2∗ , and then we may still build a new block W = V ⊕ V ∗ and apply the previous results. We assume now that neither R1 ∼ = R2 nor R1 ∼ = R2∗ , and consider the expanded block W = W1 ⊕ W2 . Recall that W1 and W2 are in the Nambu space W which decomposes as a direct sum of nonisomorphic representation spaces that are orthogonal with respect to both the unitary structure and the canonical symmetric form. Thus the decomposition W = W1 ⊕ W2 is orthogonal with respect to both of these structures. Under the assumption at hand it is immediate that EndG0 (W ) = EndG0 (W1 ) ⊕ EndG0 (W2 ) . Thus we are in a position to apply the results of Sect. 3. To do so in the case where R1 ∼ = R1∗ , we let ψ1 : R1 → R1∗ denote an equivariant isomorphism, and organize the notation so that P : V1 → V2 . Of course R1 and R2 are abstract representations, but we now choose realizations of them in V1 and V2 so that ψ2 := P ψ1 P −1 : R2 → R2∗ makes sense. Since P ψ1 P −1 (g(v2 )) = P (ψ1 (a −1 (g)P −1 (v2 ))) = P (a −1 (g)ψ1 (P −1 (v2 ))) = g(P ψ1 P −1 (v2 )) , it follows that ψ2 : R2 → R2∗ is a G0 -equivariant isomorphism. Assume for simplicity that ψ1 is symmetric, i.e., that ψ1 (v1 )(v˜1 ) = ψ1 (v˜1 )(v1 ). Then ψ2 (v2 )(v˜2 ) = P ψ1 P −1 (v2 )(v˜2 ) = ψ1 (P −1 (v2 ))(P −1 (v˜2 )) = ψ1 (P −1 (v˜2 ))(P −1 (v2 )) = P ψ1 P −1 (v˜2 )(v2 ) = ψ2 (v˜2 )(v2 ) . The computation in the case where ψ1 is odd is the same except for a sign change. Thus ψ1 and ψ2 have the same parity. Now let Ei (for i = 1, 2) be the relevant space of endomorphisms that was produced by our analysis of Wi in Sect. 3. Recall that this is either the space End(Hi ) ⊕ End(Hi∗ ) or End(Hi ⊕ Hi∗ ). Let gi be the Lie algebra of the group of unitary transformations which preserve bi . The key points now are that the unitary structure on E := E1 ⊕ E2 is the direct sum structure, the complex bilinear form on E is b = b1 ⊕ b2 , and the parity of b1 is the same as that of b2 . Thus g1 ∼ = g2 .
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For the statement of our main result in this case, let us recall that the infinitesimal versions of symmetric spaces of type II are of the form g ⊕ g/g, where the isotropy algebra is embedded diagonally. Proposition 4.6. If R1 is neither isomorphic to R2 nor to R2∗ , then the infinitesimal symmetric space associated to the T -invariant block data is a type-II ACD-symmetric space of compact type. Specifically, the classical Lie algebras sun , so2n (R), and usp2n arise in this way. Proof. Identify g1 and g2 by the isomorphism P . Call the resulting Lie algebra g. The transformations that commute with T are those in the diagonal in g ⊕ g. Thus the associated infinitesimal version of the symmetric space is of type II. The fact that the only Lie algebras which occur are those in the statement has been proved in 4.2. This completes the proof of Theorem 4.1. In closing we underline that under the assumptions of Prop. 4.6 the odd-dimensional orthogonal Lie algebra does not appear as a type-II space; only the even-dimensional one does. 5. Classification: Two Distinguished Time-Reversal Symmetries Here we describe in detail the situation where both of the distinguished time-reversal operators T and T1 are present. As would be expected, there are quite a few cases. The work will be carried out in a way which is analogous to our treatment of the case where only one time-reversal operator was present. In the first part (Sect. 5.1) we operate under the assumption that the initial truncated space V ⊕ V ∗ is invariant under both of the distinguished operators. In the second part (Sect. 5.2) we handle the general case where bigger blocks must be considered. 5.1. The case where V ⊕ V ∗ is G-invariant. Throughout, T is mixing, T1 is nonmixing and P := T T1 . Our strategy in Sects. 5.1.3 and 5.1.4 will be to first compute the operators which are b-isometries, are unitary and commute with P . This determines the Lie algebra g and its action on V ⊕ V ∗ . Then k is determined as the subalgebra of operators which commute with T or T1 , whichever is most convenient for the proof. The space of Hamiltonians is identified with g/k as before. In the case of E = End(V ) ⊕ End(V ∗ ), where g acts diagonally, the answer for g/k does not depend on the involutory properties of C, T , and T1 individually, but only on those of the nonmixing operators CP = CT T1 and T1 . The pertinent Sects. 5.1.1 and 5.1.2 are organized accordingly. 5.1.1. The case E = End(V ) ⊕ End(V ∗ ), (CP )2 = Id Recall that in the case of E = End(V ) ⊕ End(V ∗ ) it follows that the b-isometry group is SLC (V ) acting diagonally. Thus the Lie algebra g consists of those elements of the unitary algebra su(V ) which commute with the mixing unitary symmetry P . Equivalently, g is the subalgebra of su(V ) defined by commutation with the antiunitary operator CP . In the present case CP defines a real structure on V , and we have the g-invariant decomposition V = VR ⊕ iVR . Since the unitary structure , is compatible with this real structure, it follows that g = so(VR ). Our argumentation is based around T1 . If it anticommutes with P , then we replace P by iP so that it commutes. Of course this has the effect of changing to the case (CP )2 = −Id which is, however, handled below. Hence, in both cases we may assume that P and T1 commute.
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1. T12 = Id: son /(sop ⊕ soq ). The space of CP -real points VR is T1 -invariant and splits into a sum VR+ ⊕ VR− of T1 - eigenspaces. The Lie algebra k is the stabilizer of this decomposition, which is , -orthogonal. Thus k = so(VR+ ) ⊕ so(VR− ). Observe that in this case n can be any even or odd number and that p and q are arbitrary with the condition that n = p + q. 2. T12 = −Id: so2n /un . In this case T1 is a complex structure on VR which is compatible with the unitary structure. Thus k = u(VR , T1 ) and the desired result follows with 2n = dimC V . 5.1.2. The case E = End(V ) ⊕ End(V ∗ ), (CP )2 = −Id The first remarks made at the beginning of Sect. 5.1.1 still apply: g is the subalgebra of the diagonally acting su(V ) which commutes with the antiunitary operator CP . But now CP defines a C-bilinear symplectic structure on W = V ⊕ V ∗ by a(w1 , w2 ) := CP w1 , w2 . Actually CP is already defined on V and transported to V ∗ by C. Thus g = usp(V ). 1. T12 = −Id: usp2n /(usp2p ⊕ usp2q ). In this case := CT : V → V is a unitary operator which satisfies 2 = Id, and which defines the eigenspace decomposition V = V + ⊕ V − . This decomposition is both a- and , -orthogonal, and consequently k = usp(V + ) ⊕ usp(V − ). Note that there is no condition on p and q other than p + q = n. 2. T12 = Id: usp2n /un . Let VR be the T1 -real points of V . Then k is the stabilizer of VR in g = usp(V ). Here the symplectic structure a on V restricts to a real symplectic structure aR on VR . Since the unitary structure , is compatible with this structure, k is the maximal compact subalgebra un of the associated real symplectic algebra. 5.1.3. The case E = End(V ⊕ V ∗ ), b = s Recall that in this case C 2 = Id, and the b-isometry group of W = V ⊕ V ∗ is SO(W ). Before going into the various cases, let us remark on the relevance of whether or not time-reversal operators commute with P . If P 2 = u2 Id, where either u = ±1 or u = ±i, we consider the P -eigenspace decomposition W = Wu ⊕ W−u . Note dim Wu = dim W−u from Tr P = 0. The Lie algebra g ⊂ so(W ) of operators which preserve b = s and commute with P is soR (Wu ) ⊕ soR (W−u ). An antiunitary operator which commutes with P preserves the decomposition W = Wu ⊕ W−u if u = ±1, and exchanges the summands if u = ±i. Similarly, if it anticommutes with P , then it exchanges the summands in W = W+1 ⊕ W−1 and preserves the decomposition W = W+i ⊕ W−i . For this reason, as will be clear from the first case below, the sign of T T1 = ±T1 T has no bearing on our classification. 1. T 2 = T12 = Id: (son /(sop ⊕ soq )) ⊕ (son /(sop ⊕ soq )). Suppose first that P 2 = Id, giving the P -eigenspace decomposition W = W+1 ⊕ W−1 . Each of the time-reversal operators commutes with P . To determine k we consider the unitary operator = CT1 which is a mixing b-isometry satisfying P = P and 2 = Id. Thus W+1 further decomposes into a direct sum W+1 = +1 −1 ⊕ W+1 of -eigenspaces, which are orthogonal with respect to both b and , . W+1 The same discussion holds for W−1 . The stabilizer of this refined decomposition is +1 −1 +1 −1 k = soR (W+1 ) ⊕ soR (W+1 ) ⊕ soR (W−1 ) ⊕ soR (W−1 ) . From Tr P = Tr = 0 +1 −1 −1 +1 = q. one infers dim W+1 = dim W−1 = p and dim W+1 = dim W−1
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Now consider the case where P 2 = −Id but the time-reversal operators anticommute with each other and hence with P . In this situation the P -eigenspace decomposition W = W+i ⊕ W−i is still T -invariant. Therefore we are in exactly the same situation as above, and of course obtain the same result. This happens in all cases below. Thus, for the remainder of this section we assume that the time-reversal operators commute with P . 2. T 2 = T12 = −Id: (so2n /(son ⊕ son )) ⊕ (so2n /(son ⊕ son )). The situation is exactly the same as that above, except that = CT1 now satisfies 2 = −Id. Since preserves the sets of C-real points of W+1 and W−1 , defines a complex structure of these real vector spaces. Therefore we have the additional +i −i condition dim W+1 = dim W+1 on the dimensions of the -eigenspaces. 3. T 2 = −T12 : (son ⊕ son )/son . The argument to be given is true independent of whether T 2 = Id or T 2 = −Id. As usual we consider the P -eigenspace decomposition W = W+i ⊕ W−i . Since P is an isometry of both b and , , the decomposition is b- and , -orthogonal. Thus g = soR (W+i ) ⊕ soR (W−i ). Now T is antilinear and commutes with P . Thus it permutes the P -eigenspaces, i.e., T : W+i → W−i . Since k consists of those operators in g that commute with T , and T is compatible with both the unitary structure and the bilinear form b, it follows that (A, B) ∈ g is in k if and only if B = T AT −1 . In other words, after applying the obvious automorphism, k is the diagonal in g ∼ = son ⊕ son . 5.1.4. The case E = End(V ⊕ V ∗ ), b = a Recall that in this case C 2 = −Id, and the b-isometry group of W = V ⊕ V ∗ is Sp(W ). For the same reasons as indicated above we may assume that the time-reversal operators commute with P . 1. T 2 = T12 = Id: (usp2n /un ) ⊕ (usp2n /un ). Observe that the P -eigenspace decomposition W = W+1 ⊕ W−1 is a- and , orthogonal and that therefore g = usp(W+1 ) ⊕ usp(W−1 ). Let the dimension be denoted by dimC (W+1 ) = dimC (W−1 ) = 2n. Now T defines real structures on W+1 and W−1 , and these are compatible with a. R of fixed points of T is a real Hence in both cases the restriction aR to the set W±1 symplectic structure. The algebra k consists of the pairs (A, B) of operators in g R ⊕ W R . This means that A, e.g., is in the maximal compact which stabilize W+1 −1 R , i.e., in a subalgebra of the real symplectic Lie algebra determined by aR on W+1 unitary Lie algebra isomorphic to un . A similar statement holds for B. 2. T 2 = T12 = −Id: (usp2n /(usp2p ⊕ usp2q ) ⊕ (usp2n /(usp2p ⊕ usp2q )). The argument made above still shows that g = usp(W+1 ) ⊕ usp(W−1 ). Now, to determine k we consider the operator := CT1 which stabilizes this decomposition and satisfies 2 = Id. Thus the further condition to be satisfied in order for an operator to be in k is that the -eigenspace decomposition of each summand must be stabilized, i.e., k = ⊕ε,δ=±1 usp(Wεδ ). The dimensions must match pairwise because Tr P = Tr = 0. 3. T 2 = −T12 : sun /son . The answer for g/k is the same for the two cases T 2 = Id or T 2 = −Id. In either case it follows from a(w1 , w2 ) = a(P w1 , P w2 ) that the summands of the P -decomposition W = W+i ⊕ W−i are a-Lagrangian. Thus an a-isometry stabilizes the decomposition if and only if it is a C-linear transformation acting diagonally, and consequently g = su(W+i ) (which is acting diagonally as well).
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Without loss of generality we may assume that T 2 = Id (or else we replace T by T1 ). Then T is a real structure which permutes the P -eigenspaces. Thus the diagonal action (w + , w− ) → (Bw + , Bw− ) commutes with T if and only if T BT −1 = B. Since T is compatible with the initial unitary structure, it follows that B is in the associated real orthogonal group. For example, if unitary coordinates are chosen so ¯ that T is given by (z, w) → (w, ¯ z¯ ), then T BT −1 = B simply means that B = B. 5.2. Building bigger blocks. Before G0 -reduction we must determine the basic block associated to the G0 -representation space V . This has been adequately discussed in all cases with the exception of the one where there are two time-reversal operators. Here we handle that case by reducing it to the situation where there is only one. Write the initial block as V1 ⊕ V1∗ and build a diagram consisting of the four spaces Vi ⊕ Vi∗ , i = 1, . . . , 4, with the maps T , T1 , and P emanating from each of them. To be concrete, T : V1 ⊕ V1∗ → V2 ⊕ V2∗ defines V2 , and T1 : V1 ⊕ V1∗ → V3 ⊕ V3∗ defines V3 , and T1 : V2 ⊕ V2∗ → V4 ⊕ V4∗ defines V4 . The relation P = T T1 defines the remaining maps. At this point there is no need to discuss mixing. We also underline that, by the nature of the basic model, any two spaces Vi ⊕ Vi∗ and Vj ⊕ Vj∗ are either disjoint in the big Nambu space or are equal. Let us now complete the proof of our classification result, Theorem 1.1, by running through the various cases which occur in the present setting where the initial block must be extended. We only sketch this, because given how the extended block case was handled in the setting of one distinguished time-reversal symmetry (Sect. 4.4) and the detailed classification results above, the proof requires no new ideas or methods. 1) V1∗ ⊕ V1∗ is T -invariant and is not T1 -invariant. Here it is only necessary to consider P : W1 = V1 ⊕ V1∗ → V3 ⊕ V3∗ = W3 . If g is the Lie algebra of unitary operators which commute with the G0 -action and respect the b-structure on V1 ⊕ V1∗ , then the further condition of compatibility with P means that the algebra in the present case is g acting diagonally via P on W1 ⊕ W3 . Thus we have reduced to the case of only one time-reversal operator on W1 , which has been classified above. Note that this argument has nothing to do with whether or not T is mixing. Hence, in this and all of the following cases there is no need to differentiate between T and T1 . 2) V1∗ ⊕ V1∗ is neither T - nor T1 -invariant. Consider the diagram introduced above where all the spaces Wi = Vi ⊕ Vi∗ occur. If any of the Wi is invariant by either T or T1 , then we change our perspective, replace W1 by that space and apply the above argument. Thus we may assume that no Wi is stabilized by either T or T1 . It is still possible, however, that W1 = W4 , and in that case it follows that W2 = W3 . 2.1) W1 = W4 . Here both W1 and W4 are P -invariant. We leave it to the reader to check that P can be transferred to the level of End(H ) ⊕ End(H ∗ ) or End(H ⊕ H ∗ ) just as we transferred the time-reversal operators. Thus, e.g., it is enough to know the Lie algebra of operators g on W1 which are compatible with the unitary structure, are b-isometries and are compatible with P . This has been computed in Sect. 5.1. Of course we did this in the case where V ⊕ V ∗ is T - and T1 -invariant, but the compatibility with P had nothing to do with time reversal. In the present case both T and T1 exchange W1 and W2 . Thus our symmetric space is (g ⊕ g)/g.
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2.2) The spaces Wi are pairwise disjoint. Here we will go through a number of subcases, depending on whether or not there exist (equivariant) isomorphisms between various spaces. Such an isomorphism is of course assumed to be unitary and to commute with C; in particular it is a b-isometry. 2.2.1) W1 ∼ = W4 . If ϕ is the isomorphism which does this, then T ϕT −1 =: ψ is an isomorphism of W2 and W3 . Using these isomorphisms, we build W := W1 ⊕ W4 and W˜ := W2 ⊕ W3 which are of our initial type; they are stabilized by P and exchanged by T . Thus, as in 2.1, if g is the Lie algebra of operators on W which are compatible with the unitary structure, are b-isometries and are compatible with P , then our symmetric space is (g ⊕ g)/g. 2.2.2) W1 ∼ = W2 . For the reasons given above, W3 ∼ = W4 and we build W and W˜ as in that case. In the present situation P exchanges W and W˜ . We must then consider two subcases during our procedure for identifying g. The simplest case is where W and W˜ are not isomorphic. In that setting the Lie algebra g of unitary operators on W which commute with the G0 -action and are compatible with b acts diagonally on W ⊕ W˜ . This is exactly our algebra of interest. Thus in this case we can forget W˜ , and regard g as acting on W . Here T stabilizes W and thus the associated symmetric space is g/k, where k consists of the operators in g which commute with T . This situation has been classified above; in particular, only classical irreducible symmetric spaces of compact type occur. Our final case occurs under the assumption W1 ∼ = W2 in the situation where W and W˜ are isomorphic. Here we view an operator which commutes with the G0 -action as a matrix AB . CD Compatibility with P can then be interpreted as B and D being determined from A and C by P -conjugation. In this notation A : W → W and C : W → W˜ . But we may also regard C as an operator on W which is transferred to a map from W to W˜ by the isomorphism at hand. Therefore the Lie algebra of interest can be identified with the set of pairs (A, C) of operators on W which are compatible with the unitary and b-structures and commute with the G0 -action on W . Hence the associated symmetric space is the direct sum g/k ⊕ g/k, where k is determined by compatibility with T : W → W , i.e., a direct sum of two copies of an arbitrary example that occurs with only one T -symmetry. 6. Physical Realizations We now illustrate Theorem 1.1 by the two large sets of examples that were already referred to in Sect. 2: (i) fermionic quasiparticle excitations in disordered normal- and superconducting systems, and (ii) Dirac fermions in a stochastic gauge field background. In each case we fix a specific Nambu space W, and show how a variety of symmetric spaces (each corresponding to a symmetry class) is realized by varying the group of unitary and antiunitary symmetries, G. The invariable nature of W is a principle imposed by physics: electrons, e.g., have electric charge e = −1 and spin S = 1/2 and these properties cannot ever be changed. What can be changed, however, by varying the experimental conditions, are the symmetries of the Hamiltonian governing the specific situation at hand. For example, turning on an external magnetic field breaks time-reversal symmetry, adding spin-orbit scatterers to
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the system breaks spin-rotation symmetry, lowering the temperature enhances the pairing forces that may lead to a spontaneous breakdown of the global U1 charge symmetry, and so on.
6.1. Quasiparticles in metals and superconductors. The setting here is the one already described in Sect. 2.1: given the complex Hilbert space V of single-electron states, we form the Nambu space W = V ⊕ V ∗ of electron field operators. On W we then have the canonical symmetric bilinear form b, the particle-hole conjugation operator C : W → W, and the canonical unitary structure , . The complex Hilbert spaces V and V ∗ are to be viewed as representation spaces of a U1 group, which is the global U1 gauge degree of freedom of electrodynamics. Indeed, creating or annihilating one electron amounts to adding one unit of negative or positive electric charge to the fermion system. In representation-theoretic terms, this means that V carries the fundamental representation of the U1 gauge group while V ∗ carries the antifundamental one. Thus z ∈ U1 here acts on V by multiplication with z, and on V ∗ by multiplication with z¯ . Extra structure arises from the fact that electrons carry spin 1/2, which implies that V is a tensor product of spinor space, C2 , with the Hilbert space X for the orbital motion in real space. The spin-rotation group Spin3 = SU2 acts trivially on X and by the spinor representation on the factor C2 . (In a framework more comprehensive than is of relevance to the disordered systems setting developed here, the spinor representation would enter as a projective representation of the rotation group SO3 , and SO3 would act on the factor X by rotations in the three-dimensional Euclidean space.) On physical grounds, spin rotations must preserve the canonical anticommutation relations as well as the unitary structure of V. Therefore, by Prop. 2.2 spin rotations commute with the particle-hole conjugation operator C. Another symmetry operation of importance for present purposes is time reversal. As always in quantum mechanics, time reversal is implemented as an antiunitary operator T on the single-electron Hilbert space V. Its algebraic properties are influenced by the spin 1/2 nature of the electron: fundamental physics considerations dictate T 2 = −Id. A closely related condition is that time reversal commutes with spin rotations. T extends to an operation on W by CT = T C. In physics one uses the word quasiparticle for the excitations that are created by acting with a fermionic field operator on a many-fermion ground state. 6.1.1. Class D. In the general context of quasiparticle excitations in metals and superconductors, this is the fundamental class where no symmetries are present. A concrete realization takes place in superconductors where the order parameter transforms under spin rotations as a spin triplet, S = 1 (i.e., the adjoint representation of SU2 ), and transforms under SO2 -rotations of two-dimensional space as a p-wave (the fundamental representation of SO2 ). A recent candidate for a quasi-2d (or layered) spin-triplet p-wave superconductor is the compound Sr 2 Ru O4 [M, E]. (A noncharged analog is the A-phase of superfluid 3 He [VW].) Time-reversal symmetry in such a system may be broken spontaneously, or else can be broken by an external magnetic field creating vortices in the superconductor. Further realizations proposed in the recent literature include double-layer fractional quantum Hall systems at half filling [R] (more precisely, a mean-field description for the composite fermions of such systems), and a network model for the random-bond Ising model [S2].
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The time-evolution operators U = e−itH / in this class are constrained only by the requirement that they preserve both the unitary structure and the symmetric bilinear form of W. If WR is the set of real points {v + Cv : v ∈ V}, we know from Prop. 4.5 that the space of time evolutions is a real orthogonal group SO(WR ). In Cartan’s notation this is called a symmetric space of the D family. The Hamiltonians H are such that iH ∈ so(WR ); this means that the Hamiltonian matrices are imaginary skew in a suitably chosen basis (called Majorana fermions in physics). Note that since WR is a real form of (X ⊗ C2 ) ⊕ (X ⊗ C2 )∗ , the dimension of WR must be a multiple of four (for spinless particles it would only be a multiple of two). 6.1.2. Class DIII. Let now time reversal be a symmetry of the quasiparticle system. This means that magnetic fields and scattering by magnetic impurities are absent. On the other hand, spin-rotation invariance is again required to be broken. Known realizations of this situation exist in gapless superconductors, say with spinsinglet pairing, but with a sufficient concentration of spin-orbit impurities to cause strong spin-orbit scattering [S2]. In order for quasiparticle excitations to exist at low energy, the spatial symmetry of the order parameter should be d-wave (more precisely, a time-reversal invariant combination of the angular momentum l = +2 and l = −2 representations of SO2 ). A noncharged realization occurs in the B-phase of 3 He [VW], where the order parameter is spin-triplet without breaking time-reversal symmetry. Another candidate is heavy-fermion superconductors [S], where spin-orbit scattering often happens to be strong owing to the presence of elements with large atomic weights such as uranium and cerium. Time-reversal invariance constrains the set of good Hamiltonians H by H = T H T −1 . Since T 2 = −Id for spin 1/2 particles, we are dealing with the case treated in 4.3.2.1. The space of time evolutions therefore is SO(WR )/U(V), which is a symmetric space of the DIII family. The standard form of the Hamiltonians in this class is 0 Z H = , (6.1) Z∗ 0 where Z ∈ Hom(V ∗ , V) is skew. (Note again that the dimension of WR is a multiple of four, and would be a multiple of two for particles with spin zero). 6.1.3. Class C. Next let the spin of the quasiparticles be conserved, and let time-reversal symmetry be broken instead. Thus magnetic fields (or some equivalent T -breaking agent) are now present, while the effect of spin-orbit scattering is absent. The symmetry group of the physical system then is G = G0 = Spin3 = SU2 . This situation is realized in spin-singlet superconductors in the vortex phase [S4]. Prominent examples are the cuprate (or high-Tc ) superconductors [T], which are layered and exhibit d-wave symmetry in their copper-oxide planes. It has been speculated that some of these superconductors break time-reversal symmetry spontaneously, by the generation of an order-parameter component idxy or is [S3]. Other realizations of this class include network models of the spin quantum Hall effect [G]. Following the general strategy of Sect. 3, we eliminate G0 = SU2 from the picture by transferring from V ⊕ V ∗ to the reduced space X ⊕ X∗ . In the process the bilinear form b undergoes a change of parity. To see this let R = C2 (a.k.a. spinor space) be the fundamental representation space of SU2 . R is isomorphic to R ∗ by ψ : r → iσ2 r¯ , ·R , where σ2 is the second Pauli matrix. This isomorphism ψ : R → R ∗ is alternating.
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Therefore, by Prop. 3.12 the symmetric bilinear form of V ⊕ V ∗ gets transferred to the alternating form a of X ⊕ X∗ . From Prop. 4.5 we then infer that the space of time evolutions is USp(X ⊕ X∗ ) — a symmetric space of the C family. The standard form of the Hamiltonians here is A B H = , B ∗ −At with self-adjoint A ∈ End(X) and complex symmetric B ∈ Hom(X∗ , X). 6.1.4. Class CI. The next class is obtained by taking spin rotations as well as the time reversal T to be symmetries of the quasiparticle system. Thus the symmetry group is G = G0 ∪ T G0 with G0 = Spin3 = SU2 . Like in the previous symmetry class, physical realizations are provided by the lowenergy quasiparticles of unconventional spin-singlet superconductors [T]. The difference is that the superconductor must now be in the Meissner phase where magnetic field are expelled by screening currents. In the case of superconductors with several low-energy points in the first Brillouin zone, scattering off hard impurities is needed to break additional conservation laws that would otherwise emerge (see Sect. 6.1.5). To identify the relevant symmetric space, we again transfer from V ⊕ V ∗ to the reduced space X ⊕ X∗ . As before, the bilinear form b changes parity from symmetric to alternating under this reduction. In addition now, time reversal has to be transferred. As was explained in the example following Lemma 3.11, the time-reversal operator changes its involutory character from T 2 = −IdV ⊕V ∗ to T 2 = +IdX⊕X∗ . In the language of Sect. 4 the block data are V = X, E = End(V ⊕ V ∗ ), b = a, T nonmixing, and T 2 = Id. This case was treated in 4.3.3.3. From there, we know that the space of time evolutions is USp(X ⊕ X ∗ )/U(X) – a symmetric space in the CI family. The standard form of the Hamiltonians in this class is the same as that given in (6.1) but now with Z ∈ Hom(X ∗ , X) complex symmetric. 6.1.5. Class AIII. This class is commonly associated with random-matrix models for the low-energy Dirac spectrum of quantum chromodynamics with massless quarks (see Sect. 6.2.1). Here we review an alternative realization, which has recently been identified [A3] in d-wave superconductors with soft impurity scattering. To construct this realization one starts from class CI, i.e. from quasiparticles in a superconductor with time-reversal invariance and conserved spin, and enlarges the symmetry group by imposing another U1 symmetry, generated by a Hermitian operator Q with Q2 = Id. The physical reason for the extra conservation law is approximate momentum conservation in a disordered quasiparticle system with a dispersion law that has Dirac-type low-energy points at four distinct places in the Brillouin zone. Thus beyond the spin-rotation group SU2 there now exists a one-parameter group of unitary symmetries eiθQ . The operators eiθQ are defined on V, and are diagonally extended to W = V ⊕ V ∗ . They are characterized by the property that they commute with particle-hole conjugation C, time reversal T , and the spin rotations g ∈ SU2 . The reduction to standard block data is done in two steps. In the first step, we eliminate the spin-rotation group SU2 . From the previous section, the transferred data are known to be E = End(X ⊕ X∗ ), b = a, T nonmixing, and T 2 = Id. The second step is to reduce by the U1 group generated by Q. For this consider the Clinear operator J := iQ with J 2 = −Id, and let the J -eigenspace decomposition of X be written X = X+i ⊕ X−i . There is a corresponding decomposition X ∗ = X∗ +i ⊕ X∗ −i .
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Since J commutes with T , a complex structure is defined by it on the set of T -real points of X. Therefore dim X+i = dim X−i . Another consequence of J T = T J is that the C-antilinear operator T exchanges X+i with X−i . Thus T is mixing with respect to the decompositions X = X+i ⊕ X−i and X ∗ = X ∗ +i ⊕ X ∗ −i . The C-antilinear operator C maps X±i to X ∗ ∓i . The fully reduced block data now are V := X+i ⊕ X ∗ +i , E = End(V ) ⊕ End(V ∗ ), b = a, T mixing, and T 2 = Id. The finite-dimensional version of this case was treated in 4.3.1.3. Our answer for the space of time-evolution operators was SUp+q /S(Up × Uq ), which is a symmetric space in the AIII family. Unlike the general case handled in 4.3.1.3, it here follows from the fundamental physics definition of particle-hole conjugation C and time reversal T that the operator CT stabilizes a real subspace VR . We also have (CT )2 = −Id. Therefore, the operator CT defines a complex structure of VR , and hence the integers p and q, which are the dimensions of the CT -eigenspaces in V , must be equal. 6.1.6. Class A. At this point a new symmetry requirement is brought into play: conservation of the electric charge. Thus the global U1 gauge transformations of electrodynamics are now decreed to be symmetries of the quasiparticle system. This means that the system no longer is a superconductor, where U1 gauge symmetry is spontaneously broken, but is a metal or normal-conducting system. If all further symmetries are broken (time reversal by a magnetic field or magnetic impurities, spin rotations by spin-orbit scattering, etc.), the symmetry group is G = G0 = U1 . All states (actually, field operators) in V have the same electric charge. Thus the irreducible U1 representations which they carry all have the same isomorphism class, say λ. States in V ∗ carry the opposite charge and belong to the dual class λ∗ . Since λ = λ∗ , we are in the situation of Sect. 4.3.1, where E = End(V) ⊕ End(V ∗ ). With T being absent, the space of time evolutions is U(V) acting diagonally on V ⊕ V ∗ . In random-matrix theory, and in the finite-dimensional case where U(V) ∼ = UN , one refers to these matrix spaces as the circular Wigner-Dyson class of unitary symmetry. The Hamiltonians in this class are represented by complex Hermitian matrices. If we make the restriction to traceless Hamiltonians, the space of time evolutions becomes SUN , which is a type-II irreducible symmetric space of the A family. 6.1.7. Class AII. Beyond charge conservation or U1 gauge symmetry, time reversal T is now required to be a symmetry of the quasiparticle system. Physical realizations of this case occur in metallic systems with spin-orbit scattering. The pioneering experimental work (of the weak localization phenomenon in this class) was done on disordered magnesium films with gold impurities [B]. The block data now is E = End(V) ⊕ End(V ∗ ), b = s, T nonmixing, T 2 = −Id. This case was considered in 4.3.1.1. The main point there was that time reversal T defines a C-linear symplectic structure a on V by a(v1 , v2 ) = T v1 , v2 . Conjugation by T therefore fixes a unitary symplectic group USp(V) inside of U(V), and the space of good time evolutions is G/K = U(V)/USp(V). In the finite-dimensional setting where G/K ∼ = U2N /USp2N , this is called the circular Wigner-Dyson class of symplectic symmetry in random-matrix theory. The Hamiltonians in this class are represented by Hermitian matrices whose matrix entries are real quaternions. The irreducible part SU2N /USp2N , obtained by restricting to traceless Hamiltonians, is a type-I symmetric space in the AII family.
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6.1.8. Class AI. The next class is the Wigner-Dyson class of orthogonal symmetry. In the present quasiparticle setting it is obtained by imposing spin-rotation symmetry, U1 gauge (or charge) symmetry and time-reversal symmetry all at once. Important physical realizations are by disordered metals in zero magnetic field. Families of quantum chaotic billiards also belong to this class. The group of unitary symmetries here is G0 = U1 × SU2 . We eliminate the spinrotation group SU2 from the picture by transferring from V = X ⊗ C2 to the reduced space X. Again, the involutory character of T is reversed in the process: the transferred time reversal satisfies T 2 = +Id. The parity of the bilinear form also changes, from symmetric to alternating; however, this turns out to be irrelevant here, as there is still the U1 charge symmetry and we are in the situation λ = λ∗ . The block data now is E = End(X) ⊕ End(X∗ ), b = a, T nonmixing, T 2 = Id. According to 4.3.1.2 these yield (the Cartan embedding of) U(X)/O(X) as the space of good time evolutions. The irreducible part SU(X)/SO(X), or SUN /SON in the finitedimensional setting, is a symmetric space in the AI family. The Hamiltonian matrices in this class can be arranged to be real symmetric.
6.2. The Euclidean Dirac operator for chiral fermions. We now explore the physical examples afforded by Dirac fermions in a random gauge field background. These examples include the Dirac operator of quantum chromodynamics, i.e., the theory of strong SU3 gauge interactions between elementary particles called quarks. The mathematical setting for this has already been described in Sect. 2.3. Recall that one is given a twisted spinor bundle S ⊗ R over Euclidean space-time, and that V is taken to be the Hilbert space of L2 -sections of that bundle. One is interested in the Dirac operator DA in a gauge field background A and in the limit of zero mass: DA = iγ µ (∂µ − Aµ ) . We extend the self-adjoint operator DA diagonally from V to the fermionic Nambu space W = V ⊕V ∗ by the condition DA = −CDA C −1 . The chiral ‘symmetry’DA +DA = 0, where = γ5 is the chirality operator, then becomes a true symmetry DA = T DA T −1 with an antiunitary operator T = C = C, which mixes V and V ∗ . 6.2.1. Class AIII. Let now the complex vector space R = CN be the fundamental representation space for the gauge group SUN with N ≥ 3. (N is called the number of colors in this context.) Quantum chromodynamics is the special case N = 3. The fact that the extended Dirac operator DA acts diagonally on W = V ⊕ V ∗ is attributed to a symmetry group G0 = U1 which has V and V ∗ as inequivalent representation spaces. For a generic gauge-field configuration there exist no further symmetries; thus the total symmetry group is G = G0 ∪ T G0 . The block data here is V = V, E = End(V ) ⊕ End(V ∗ ), b = s, T mixing, T 2 = Id, which is the case considered in 4.3.1.3. If n = dim V , we have p∼ = sun /s(up ⊕ uq ) . The difference of integers p − q is to be identified with the difference between the num2 . (‘Right’ and ‘left’ in this context pertain to the ber of right and left zero modes of DA (+1)- and (−1)-eigenspaces of the chirality = γ5 .) The latter number is a topological invariant called the index of the Dirac operator.
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6.2.2. Class BDI. We retain the framework from before, but now consider the gauge group SU2 , where the number of colors N = 2. In this case the massless Dirac operator DA has an additional antiunitary symmetry [V1], which emerges as follows. Recall that the unitary SU2 -representation space R = C2 is isomorphic to the dual representation space R ∗ by a C-linear mapping ψ : R → R ∗ . Combining the inverse of this with ι : R → R ∗ defined by ι(r) = r, ·R , we obtain a C-antilinear mapping β := ψ −1 ◦ ι : R → R. The map β thus defined commutes with the SU2 -action on R. By Lemma 3.11 it satisfies β 2 = −IdR since ψ is alternating. Now, on the (untwisted) spinor bundle S over Euclidean space-time M there exists a C-antilinear operator α, called charge conjugation in physics, which anticommutes with the Clifford action γ : T ∗ M → End(S); thus αiγ = iγ α. Since γ5 = γ 0 γ 1 γ 2 γ 3 , this implies that α commutes with γ5 = and stabilizes the -eigenspace decomposition S = S+ ⊕ S− into half-spinor components S± . The charge conjugation operator has square α 2 = −IdS . For the case of three or more colors, the existence of α is of no consequence from a symmetry perspective, as the fundamental and antifundamental representations of SUN are inequivalent for N ≥ 3. For N = 2, however, we also have β, and α combines with it to give an antiunitary symmetry T1 = α ⊗ β. Indeed, T1 DA T1−1 = (α ⊗ β)DA (α ⊗ β) = α(iγ µ )α −1 ⊗ β(∂µ − Aµ )β −1 . Since gauge transformations g(x) ∈ SU2 commute with β, so do the components Aµ (x) ∈ su2 of the gauge field. Thus βAµ β −1 = Aµ , and since α(iγ )α −1 = iγ , we have T1 DA T1−1 = DA . Note that the antiunitary symmetry T1 : V → V is nonmixing, and T12 = Id. As usual, the extension to an operator T1 : W → W is made by requiring CT1 = T1 C. Thus we now have two antiunitary symmetries, T and T1 . Because T is mixing and T1 nonmixing, the unitary operator P = T T1 = T1 T mixes V with V ∗ . Since T 2 = T12 = Id, and (CP )2 = Id, this is the case treated in 5.1.1.1, where we found p∼ = so(VR )/(so(VR+ ) ⊕ so(VR− )) . After truncation to finite dimension this is sop+q /(sop ⊕ soq ). The difference p − q still has a topological interpretation as the index of the Dirac operator. Although our considerations explicitly referred to the case of the gauge group being SU2 , the only specific feature we used was the existence of an alternating isomorphism ψ : R → R ∗ . The same result therefore holds for any gauge group representation R where such an isomorphism exists. In particular it holds for the fundamental representation of the whole series of symplectic groups USp2N (which includes SU2 ∼ = USp2 ). 6.2.3. Class CII. Now take R to be the adjoint representation of any compact Lie (gauge) group K with semisimple Lie algebra. This case is called ‘adjoint fermions’ in physics. A detailed symmetry analysis of it was presented in [H2]. The Cartan-Killing form on Lie(K), B(X, Y ) = Tr ad(X)ad(Y ), is nondegenerate, invariant, complex bilinear, and symmetric. B therefore defines an isomorphism ψ : R → R ∗ by ψ(X) = B(X, ·). Since B is symmetric, so is ψ.
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The change in parity of ψ reverses the parity of the antiunitary operator β = ψ −1 ◦ ι, which now satisfies β 2 = +IdR . By α 2 = −Id this translates to T12 = (α ⊗ β)2 = −Id. Thus we now have two antiunitary symmetries T and T1 with T 2 = Id = −T12 , and (CP )2 = (CT T1 )2 = −Id. This case was handled in 5.1.2.1 where we found p∼ = usp(V)/(usp(V + ) ⊕ usp(V − )) . In a finite-dimensional setting this would be usp2p+2q /(usp2p ⊕ usp2q ). In summary, the physical situation is ruled by a mathematical trichotomy: the isomorphism ψ : R → R ∗ is either symmetric, or alternating, or does not exist. The corresponding symmetry class of the massless Dirac operator is CII, BDI, or AIII, respectively. As was first observed by Verbaarschot [V], this is the same trichotomy that ruled Dyson’s threefold way. Acknowledgement. This work was carried out under the auspices of the Deutsche Forschungsgemeinschaft, SFB/TR12. Major portions of the article were prepared while M.R.Z. was visiting the Institute for Advanced Study (Princeton, USA) and the Newton Institute for Mathematical Sciences (Cambridge, UK). The support of these institutions is gratefully acknowledged.
References [A2] [A3] [A] [B] [B3] [C] [D] [E] [G] [H2] [H] [K] [M] [R] [S2] [S3] [S4] [S]
Altland, A., Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-/superconducting hybrid systems. Phys. Rev. B 55, 1142–1161 (1997) Altland, A., Simons, B.D., Zirnbauer, M.R.: Theories of low-energy quasiparticle states in disordered d-wave superconductors. Phys. Rep. 359, 283-354 (2002) Arnold, V.I.: Mathematical methods of classical mechanics. New York, Heidelberg, Berlin: Springer-Verlag, 1978 Bergmann, G.: Weak localization in thin films – a time-of-flight experiment with conduction electrons. Phys. Rep. 107, 1–58 (1984) Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Berlin, Heidelberg, New York: Springer-Verlag, 1992 Caselle, M., Magnea, U.: Random-matrix theory and symmetric spaces. Phys. Rep. 394, 41–156 (2004) Dyson, F.J.: The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1962) Eremin, I., Manske, D., Ovchinnikov, S.G., Annett, J.F.: Unconventional superconductivity and magnetism in Sr 2 RuO4 and related materials. Ann. Physik 13, 149–174 (2004) Gruzberg, I.A., Ludwig, A.W.W., Read, N.: Exact exponents for the spin quantum Hall transition. Phys. Rev. Lett. 82, 4524–4527 (1999) Halasz, M.A.,Verbaarschot, J.J.M.: Effective Lagrangians and chiral random-matrix theory. Phys. Rev. D 51, 2563–2573 (1995) Helgason, S.: Differential geometry, Lie groups and symmetric spaces. New York: Academic Press, 1978 Katz, N.M., Sarnak, P.: Random matrices, Frobenius eigenvalues, and monodromy. Providence, R.I.: American Mathematical Society, 1999 Mackenzie, A.P., Maeno, Y.: The superconductivity of Sr 2 RuO4 and the physics of spin-triplet pairing. Rev. Mod. Phys. 75, 657–712 (2003) Read, N., Green, D.: Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000) Senthil, T., Fisher, M.P.A.: Quasiparticle localization in superconductors with spin-orbit scattering. Phys. Rev. B 61, 9690–9698 (2000) Senthil, T., Marston, J.B., Fisher, M.P.A.: Spin quantum Hall effect in unconventional superconductors. Phys. Rev. B 60, 4245–4254 (1999) Senthil, T., Fisher, M.P.A., Balents, L., Nayak, C.: Quasiparticle transport and localization in high-Tc superconductors. Phys. Rev. Lett. 81, 4704–4707 (1998) Stewart, G.S.: Heavy-fermion systems. Rev. Mod. Phys. 56, 755–787 (1984)
Symmetry Classes of Disordered Fermions [T]
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Tsuei, C.C., Kirtley, J.R.: Pairing symmetry in the cuprate superconductors. Rev. Mod. Phys. 72, 969–1016 (2000) [V] Verbaarschot, J.J.M.: The spectrum of the QCD Dirac operator and chiral random-matrix theory: the threefold way. Phys. Rev. Lett. 72, 2531–2533 (1994) [V1] Verbaarschot, J.J.M.: The spectrum of the Dirac operator near zero virtuality for Nc = 2. Nucl. Phys. B 426, 559–574 (1994) [V2] Verbaarschot, J.J.M., Zahed, I.: Spectral density of the QCD Dirac operator near zero virtuality. Phys. Rev. Lett. 70, 3852–3855 (1993) [VW] Vollhardt, D., W¨olfle, P.: The superfluid phases of Helium 3. London: Taylor & Francis, 1990 [Z] Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986–5018 (1996) Communicated by P. Sarnak